From df4a2e1f3070983332af3d30d4ea7a6cddbb091d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 12 Jun 2026 17:30:55 -0700
Subject: [PATCH 01/26] Prove it is an Ab

---
 src/analysis.lagda.md                         |   4 +
 ...roximations-metric-abelian-groups.lagda.md | 396 +++++++++++++
 ...roximations-metric-abelian-groups.lagda.md |  70 +++
 ...completions-metric-abelian-groups.lagda.md |  81 +++
 ...ocompletion-metric-abelian-groups.lagda.md | 540 ++++++++++++++++++
 src/analysis/metric-abelian-groups.lagda.md   |  92 ++-
 ...quality-positive-rational-numbers.lagda.md |  13 +
 .../functoriality-set-quotients.lagda.md      |   3 +-
 src/metric-spaces.lagda.md                    |   3 +
 ...imations-isometries-metric-spaces.lagda.md |  72 +++
 ...imations-short-maps-metric-spaces.lagda.md |  72 +++
 ...ns-short-maps-pseudometric-spaces.lagda.md |  84 +++
 ...seudocompletions-of-metric-spaces.lagda.md |  55 ++
 ...ompletions-of-pseudometric-spaces.lagda.md |  25 +
 14 files changed, 1499 insertions(+), 11 deletions(-)
 create mode 100644 src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
 create mode 100644 src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md
 create mode 100644 src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
 create mode 100644 src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md

diff --git a/src/analysis.lagda.md b/src/analysis.lagda.md
index bce22009164..38b2145e149 100644
--- a/src/analysis.lagda.md
+++ b/src/analysis.lagda.md
@@ -3,11 +3,15 @@
 ```agda
 module analysis where
 
+open import analysis.addition-cauchy-approximations-metric-abelian-groups public
 open import analysis.alternation-sequences-metric-abelian-groups public
+open import analysis.cauchy-approximations-metric-abelian-groups public
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups public
 open import analysis.complete-metric-abelian-groups public
 open import analysis.convergent-series-complete-metric-abelian-groups public
 open import analysis.convergent-series-metric-abelian-groups public
 open import analysis.limits-of-sequences-metric-abelian-groups public
+open import analysis.metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups public
 open import analysis.metric-abelian-groups public
 open import analysis.metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups public
 open import analysis.sequences-metric-abelian-groups public
diff --git a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..340ee0f9d73
--- /dev/null
+++ b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -0,0 +1,396 @@
+# Addition of Cauchy approximations in metric abelian groups
+
+```agda
+module analysis.addition-cauchy-approximations-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.strict-inequality-positive-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.binary-functoriality-set-quotients
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.metric-quotients-of-pseudometric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+[Cauchy approximations](analysis.cauchy-approximations-metric-abelian-groups.md)
+in [metric abelian groups](analysis.metric-abelian-groups.md) admit an addition
+operation whose properties resemble an
+[abelian group](group-theory.abelian-groups.md) with respect to the
+[similarity relationship](metric-spaces.similarity-of-elements-pseudometric-spaces.md)
+of the
+[Cauchy pseudocompletion of the metric abelian group](analysis.cauchy-pseudocompletions-metric-abelian-groups.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  ((x , is-approx-x) (y , is-approx-y) :
+    cauchy-approximation-Metric-Ab G)
+  where
+
+  opaque
+    map-add-cauchy-approximation-Metric-Ab :
+      ℚ⁺ → type-Metric-Ab G
+    map-add-cauchy-approximation-Metric-Ab ε =
+      let (δ , _) = bound-double-le-ℚ⁺ ε in add-Metric-Ab G (x δ) (y δ)
+
+  abstract opaque
+    unfolding map-add-cauchy-approximation-Metric-Ab
+
+    is-cauchy-approximation-map-add-cauchy-approximation-Metric-Ab :
+      is-cauchy-approximation-Metric-Ab G map-add-cauchy-approximation-Metric-Ab
+    is-cauchy-approximation-map-add-cauchy-approximation-Metric-Ab δ ε =
+      let
+        (δ' , 2δ'<δ) = bound-double-le-ℚ⁺ δ
+        (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      in
+        monotonic-neighborhood-Metric-Ab G
+          ( add-Metric-Ab G (x δ') (y δ'))
+          ( add-Metric-Ab G (x ε') (y ε'))
+          ( (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε'))
+          ( δ +ℚ⁺ ε)
+          ( concat-eq-le-ℚ⁺
+            { z = δ +ℚ⁺ ε}
+            ( interchange-law-add-add-ℚ⁺ δ' ε' δ' ε')
+            ( preserves-le-add-ℚ 2δ'<δ 2ε'<ε))
+          ( neighborhood-add-Metric-Ab
+            ( G)
+            ( δ' +ℚ⁺ ε')
+            ( δ' +ℚ⁺ ε')
+            ( x δ')
+            ( x ε')
+            ( y δ')
+            ( y ε')
+            ( is-approx-x δ' ε')
+            ( is-approx-y δ' ε'))
+
+  add-cauchy-approximation-Metric-Ab : cauchy-approximation-Metric-Ab G
+  add-cauchy-approximation-Metric-Ab =
+    ( map-add-cauchy-approximation-Metric-Ab ,
+      is-cauchy-approximation-map-add-cauchy-approximation-Metric-Ab)
+```
+
+## Properties
+
+### Addition of Cauchy approximations is a similarity-preserving binary map
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  preserves-sim-add-cauchy-approximation-Metric-Ab :
+    preserves-sim-binary-map-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( add-cauchy-approximation-Metric-Ab G)
+  preserves-sim-add-cauchy-approximation-Metric-Ab
+    {x , is-approx-x} {x' , is-approx-x'} {y , is-approx-y} {y' , is-approx-y'}
+    x~x' y~y' δ ε θ =
+    let
+      (δ' , 2δ'<δ) = bound-double-le-ℚ⁺ δ
+      (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      (θ' , 2θ'<θ) = bound-double-le-ℚ⁺ θ
+    in
+      monotonic-neighborhood-Metric-Ab G
+        ( add-Metric-Ab G (x ε') (y ε'))
+        ( add-Metric-Ab G (x' θ') (y' θ'))
+        ( (ε' +ℚ⁺ θ' +ℚ⁺ δ') +ℚ⁺ (ε' +ℚ⁺ θ' +ℚ⁺ δ'))
+        ( ε +ℚ⁺ θ +ℚ⁺ δ)
+        ( concat-eq-le-ℚ⁺
+          { z = ε +ℚ⁺ θ +ℚ⁺ δ}
+          ( equational-reasoning
+            (ε' +ℚ⁺ θ' +ℚ⁺ δ') +ℚ⁺ (ε' +ℚ⁺ θ' +ℚ⁺ δ')
+            ＝ ((ε' +ℚ⁺ θ') +ℚ⁺ (ε' +ℚ⁺ θ')) +ℚ⁺ (δ' +ℚ⁺ δ')
+              by interchange-law-add-add-ℚ⁺ _ _ _ _
+            ＝ (ε' +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ θ') +ℚ⁺ (δ' +ℚ⁺ δ')
+              by ap-add-ℚ⁺ (interchange-law-add-add-ℚ⁺ ε' θ' ε' θ') refl)
+          ( preserves-le-add-ℚ
+            ( preserves-le-add-ℚ 2ε'<ε 2θ'<θ)
+            ( 2δ'<δ)))
+        ( neighborhood-add-Metric-Ab
+          ( G)
+          ( ε' +ℚ⁺ θ' +ℚ⁺ δ')
+          ( ε' +ℚ⁺ θ' +ℚ⁺ δ')
+          ( x ε')
+          ( x' θ')
+          ( y ε')
+          ( y' θ')
+          ( x~x' δ' ε' θ')
+          ( y~y' δ' ε' θ'))
+```
+
+### The addition of two constant Cauchy approximations for `x` and `y` is similar to the constant approximation for `x + y`
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  sim-add-const-cauchy-approximation-Metric-Ab :
+    (x y : type-Metric-Ab G) →
+    sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space
+        ( metric-space-Metric-Ab G))
+      ( add-cauchy-approximation-Metric-Ab G
+        ( const-cauchy-approximation-Metric-Ab G x)
+        ( const-cauchy-approximation-Metric-Ab G y))
+      ( const-cauchy-approximation-Metric-Ab G (add-Metric-Ab G x y))
+  sim-add-const-cauchy-approximation-Metric-Ab x y δ ε θ =
+    refl-neighborhood-Metric-Ab G (ε +ℚ⁺ θ +ℚ⁺ δ) (add-Metric-Ab G x y)
+```
+
+### Addition is associative relative to the similarity relation
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) ay@(y , is-approx-y) az@(z , is-approx-z) :
+    cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  sim-associative-add-cauchy-approximation-Metric-Ab :
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( add-cauchy-approximation-Metric-Ab G
+        ( add-cauchy-approximation-Metric-Ab G ax ay)
+        ( az))
+      ( add-cauchy-approximation-Metric-Ab G
+        ( ax)
+        ( add-cauchy-approximation-Metric-Ab G ay az))
+  sim-associative-add-cauchy-approximation-Metric-Ab δ ε θ =
+    let
+      (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      (ε'' , 2ε''<ε') = bound-double-le-ℚ⁺ ε'
+      (θ' , 2θ'<θ) = bound-double-le-ℚ⁺ θ
+      (θ'' , 2θ''<θ') = bound-double-le-ℚ⁺ θ'
+      xyz1 = add-Metric-Ab G (add-Metric-Ab G (x ε'') (y ε'')) (z ε')
+      xyz2 = add-Metric-Ab G (add-Metric-Ab G (x θ') (y θ'')) (z θ'')
+    in
+      tr
+        ( neighborhood-Metric-Ab G (ε +ℚ⁺ θ +ℚ⁺ δ) xyz1)
+        ( associative-add-Metric-Ab G _ _ _)
+        ( monotonic-neighborhood-Metric-Ab G
+          ( xyz1)
+          ( xyz2)
+          ( (ε'' +ℚ⁺ θ') +ℚ⁺ (ε'' +ℚ⁺ θ'') +ℚ⁺ (ε' +ℚ⁺ θ''))
+          ( ε +ℚ⁺ θ +ℚ⁺ δ)
+          ( concat-eq-le-ℚ⁺
+            { z = ε +ℚ⁺ θ +ℚ⁺ δ}
+            ( equational-reasoning
+              (ε'' +ℚ⁺ θ') +ℚ⁺ (ε'' +ℚ⁺ θ'') +ℚ⁺ (ε' +ℚ⁺ θ'')
+              ＝ ((ε'' +ℚ⁺ ε'') +ℚ⁺ (θ' +ℚ⁺ θ'')) +ℚ⁺ (ε' +ℚ⁺ θ'')
+                by ap-add-ℚ⁺ (interchange-law-add-add-ℚ⁺ _ _ _ _) refl
+              ＝ ((ε'' +ℚ⁺ ε'') +ℚ⁺ ε') +ℚ⁺ ((θ' +ℚ⁺ θ'') +ℚ⁺ θ'')
+                by interchange-law-add-add-ℚ⁺ _ _ _ _
+              ＝ ((ε'' +ℚ⁺ ε'') +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ (θ'' +ℚ⁺ θ''))
+                by ap-add-ℚ⁺ refl (associative-add-ℚ⁺ _ _ _))
+            ( transitive-le-ℚ⁺
+              ( ((ε'' +ℚ⁺ ε'') +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ (θ'' +ℚ⁺ θ'')))
+              ( (ε' +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ θ'))
+              ( ε +ℚ⁺ θ +ℚ⁺ δ)
+              ( transitive-le-ℚ⁺
+                ( (ε' +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ θ'))
+                ( ε +ℚ⁺ θ)
+                ( ε +ℚ⁺ θ +ℚ⁺ δ)
+                ( le-left-add-ℚ⁺ (ε +ℚ⁺ θ) δ)
+                ( preserves-le-add-ℚ 2ε'<ε 2θ'<θ))
+              ( preserves-le-add-ℚ
+                ( preserves-le-left-add-ℚ _ _ _ 2ε''<ε')
+                ( preserves-le-right-add-ℚ _ _ _ 2θ''<θ'))))
+          ( neighborhood-add-Metric-Ab G
+            ( (ε'' +ℚ⁺ θ') +ℚ⁺ (ε'' +ℚ⁺ θ''))
+            ( ε' +ℚ⁺ θ'')
+            ( add-Metric-Ab G (x ε'') (y ε''))
+            ( add-Metric-Ab G (x θ') (y θ''))
+            ( z ε')
+            ( z θ'')
+            ( neighborhood-add-Metric-Ab G
+              ( ε'' +ℚ⁺ θ')
+              ( ε'' +ℚ⁺ θ'')
+              ( x ε'')
+              ( x θ')
+              ( y ε'')
+              ( y θ'')
+              ( is-approx-x ε'' θ')
+              ( is-approx-y ε'' θ''))
+            ( is-approx-z ε' θ'')))
+```
+
+### Commutativity of addition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (x y : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  commutative-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G x y ＝
+    add-cauchy-approximation-Metric-Ab G y x
+  commutative-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → commutative-add-Metric-Ab G _ _))
+```
+
+### Unit laws
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  sim-left-unit-law-add-cauchy-approximation-Metric-Ab :
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( add-cauchy-approximation-Metric-Ab G
+        ( zero-cauchy-approximation-Metric-Ab G)
+        ( ax))
+      ( ax)
+  sim-left-unit-law-add-cauchy-approximation-Metric-Ab δ ε θ =
+    let (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε in
+    monotonic-neighborhood-Metric-Ab G
+      ( add-Metric-Ab G (zero-Metric-Ab G) (x ε'))
+      ( x θ)
+      ( ε' +ℚ⁺ θ)
+      ( ε +ℚ⁺ θ +ℚ⁺ δ)
+      ( transitive-le-ℚ⁺
+        ( ε' +ℚ⁺ θ)
+        ( ε +ℚ⁺ θ)
+        ( ε +ℚ⁺ θ +ℚ⁺ δ)
+        ( le-left-add-ℚ⁺ (ε +ℚ⁺ θ) δ)
+        ( preserves-le-left-add-ℚ _ _ _ (le-modulus-le-double-le-ℚ⁺ ε)))
+      ( inv-tr
+        ( λ y → neighborhood-Metric-Ab G (ε' +ℚ⁺ θ) y (x θ))
+        ( left-unit-law-add-Metric-Ab G (x ε'))
+        ( is-approx-x ε' θ))
+
+  sim-right-unit-law-add-cauchy-approximation-Metric-Ab :
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( add-cauchy-approximation-Metric-Ab G
+        ( ax)
+        ( zero-cauchy-approximation-Metric-Ab G))
+      ( ax)
+  sim-right-unit-law-add-cauchy-approximation-Metric-Ab =
+    tr
+      ( λ ay → sim-cauchy-pseudocompletion-Metric-Ab G ay ax)
+      ( commutative-add-cauchy-approximation-Metric-Ab G
+        ( zero-cauchy-approximation-Metric-Ab G)
+        ( ax))
+      ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab)
+```
+
+### Negations of Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  neg-cauchy-approximation-Metric-Ab :
+    cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
+  neg-cauchy-approximation-Metric-Ab =
+    map-isometry-cauchy-approximation-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-space-Metric-Ab G)
+      ( isometry-neg-Metric-Ab G)
+```
+
+### Inverse laws of Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  left-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( neg-cauchy-approximation-Metric-Ab G ax)
+      ( ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  left-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → left-inverse-law-add-Metric-Ab G _))
+
+  right-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( ax)
+      ( neg-cauchy-approximation-Metric-Ab G ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  right-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → right-inverse-law-add-Metric-Ab G _))
+```
+
+### Negations of Cauchy approximations preserve similarity
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  preserves-sim-neg-cauchy-approximation-Metric-Ab :
+    (x y : cauchy-approximation-Metric-Ab G) →
+    sim-cauchy-pseudocompletion-Metric-Ab G x y →
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( neg-cauchy-approximation-Metric-Ab G x)
+      ( neg-cauchy-approximation-Metric-Ab G y)
+  preserves-sim-neg-cauchy-approximation-Metric-Ab x y =
+    preserves-sim-isometry-cauchy-pseudocompletion-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-space-Metric-Ab G)
+      ( isometry-neg-Metric-Ab G)
+      { x}
+      { y}
+```
diff --git a/src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..f93cdc3d811
--- /dev/null
+++ b/src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md
@@ -0,0 +1,70 @@
+# Cauchy approximations in metric abelian groups
+
+```agda
+module analysis.cauchy-approximations-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-approximations-metric-spaces
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "Cauchy approximation" Disambiguation="in a metric abelian group" Agda=cauchy-approximation-Metric-Ab}}
+in a [metric abelian group](analysis.metric-abelian-groups.md) is a
+[Cauchy approximation](metric-spaces.cauchy-approximations-metric-spaces.md) in
+the underlying [metric space](metric-spaces.metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  is-cauchy-approximation-prop-Metric-Ab : subtype l2 (ℚ⁺ → type-Metric-Ab G)
+  is-cauchy-approximation-prop-Metric-Ab =
+    is-cauchy-approximation-prop-Metric-Space (metric-space-Metric-Ab G)
+
+  is-cauchy-approximation-Metric-Ab : (ℚ⁺ → type-Metric-Ab G) → UU l2
+  is-cauchy-approximation-Metric-Ab =
+    is-in-subtype is-cauchy-approximation-prop-Metric-Ab
+
+  cauchy-approximation-Metric-Ab : UU (l1 ⊔ l2)
+  cauchy-approximation-Metric-Ab =
+    type-subtype is-cauchy-approximation-prop-Metric-Ab
+```
+
+## Properties
+
+### Constant maps in metric abelian groups are Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  const-cauchy-approximation-Metric-Ab :
+    type-Metric-Ab G → cauchy-approximation-Metric-Ab G
+  const-cauchy-approximation-Metric-Ab =
+    const-cauchy-approximation-Metric-Space (metric-space-Metric-Ab G)
+
+  zero-cauchy-approximation-Metric-Ab :
+    cauchy-approximation-Metric-Ab G
+  zero-cauchy-approximation-Metric-Ab =
+    const-cauchy-approximation-Metric-Ab (zero-Metric-Ab G)
+```
diff --git a/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
new file mode 100644
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+# Cauchy pseudocompletions of metric abelian groups
+
+```agda
+module analysis.cauchy-pseudocompletions-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import foundation.binary-relations
+open import foundation.equivalence-relations
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.pseudometric-spaces
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "Cauchy pseudocompletion" Disambiguation="of a metric abelian group" Agda=cauchy-pseudocompletion-Metric-Ab}}
+of a [metric abelian group](group-theory.metric-abelian-groups.md) is the
+[Cauchy pseudocompletion](metric-spaces.cauchy-pseudocompletions-of-metric-spaces.md)
+of the underlying [metric space](metric-spaces.metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  cauchy-pseudocompletion-Metric-Ab : Pseudometric-Space (l1 ⊔ l2) l2
+  cauchy-pseudocompletion-Metric-Ab =
+    cauchy-pseudocompletion-Metric-Space (metric-space-Metric-Ab G)
+
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab :
+    equivalence-relation l2 (cauchy-approximation-Metric-Ab G)
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab =
+    equivalence-relation-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab)
+
+  sim-prop-cauchy-pseudocompletion-Metric-Ab :
+    Relation-Prop l2 (cauchy-approximation-Metric-Ab G)
+  sim-prop-cauchy-pseudocompletion-Metric-Ab =
+    sim-prop-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+
+  sim-cauchy-pseudocompletion-Metric-Ab :
+    Relation l2 (cauchy-approximation-Metric-Ab G)
+  sim-cauchy-pseudocompletion-Metric-Ab =
+    sim-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+```
+
+## Properties
+
+### If two constant Cauchy approximations are similar, they have the same constant
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab :
+    (x y : type-Metric-Ab G) →
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( const-cauchy-approximation-Metric-Ab G x)
+      ( const-cauchy-approximation-Metric-Ab G y) →
+    x ＝ y
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab =
+    eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Space
+      ( metric-space-Metric-Ab G)
+```
diff --git a/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
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+# The metric abelian group formed by the metric quotient of Cauchy pseudocompletions of metric abelian groups
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module analysis.metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.addition-cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.addition-positive-rational-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-functoriality-set-quotients
+open import foundation.dependent-pair-types
+open import foundation.embeddings
+open import foundation.equivalence-relations
+open import foundation.functoriality-set-quotients
+open import foundation.identity-types
+open import foundation.injective-maps
+open import foundation.set-quotients
+open import foundation.sets
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import group-theory.abelian-groups
+open import group-theory.groups
+open import group-theory.monoids
+open import group-theory.semigroups
+
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.metric-quotients-of-pseudometric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.pseudometric-spaces
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The [metric quotient](metric-spaces.metric-quotients-of-pseudometric-spaces.md)
+of the
+[Cauchy pseudocompletion](analysis.cauchy-pseudocompletions-metric-abelian-groups.md)
+of a [metric abelian group](analysis.metric-abelian-groups.md) itself forms a
+metric abelian group.
+
+This construction is precisely analogous to the definition of the Cauchy real
+numbers and their definition of addition.
+
+## Definition
+
+### The metric space of the metric quotient
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Metric-Space (l1 ⊔ l2) (l1 ⊔ l2)
+  metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    metric-quotient-Pseudometric-Space (cauchy-pseudocompletion-Metric-Ab G)
+
+  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Set (l1 ⊔ l2)
+  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    set-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab : UU (l1 ⊔ l2)
+  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    type-Set set-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+```
+
+### Addition in the metric quotient
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  binary-hom-add-cauchy-pseudocompletion-Metric-Ab :
+    binary-hom-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+  binary-hom-add-cauchy-pseudocompletion-Metric-Ab =
+    ( add-cauchy-approximation-Metric-Ab G ,
+      λ {x} {x'} {y} {y'} →
+        preserves-sim-add-cauchy-approximation-Metric-Ab G {x} {x'} {y} {y'})
+
+  add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    binary-map-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab)
+```
+
+### The embedding of elements of a metric abelian group in the metric quotient of its Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    cauchy-approximation-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    quotient-map
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+
+  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-Metric-Ab G → type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( const-cauchy-approximation-Metric-Ab G x)
+
+  abstract
+    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      is-injective in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      {x} {y} inx=iny =
+      eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( x)
+        ( y)
+        ( apply-effectiveness-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( inx=iny))
+
+    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      is-emb in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+      is-emb-is-injective
+        ( is-set-type-Metric-Space
+          ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+        is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-Metric-Ab G ↪ type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+
+  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab (zero-Metric-Ab G)
+```
+
+## Properties
+
+### The embedding in the metric quotient of the Cauchy pseudocompletion preserves addition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y : cauchy-approximation-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( x))
+      ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( y)) ＝
+    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( G)
+      ( add-cauchy-approximation-Metric-Ab G x y)
+  add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    compute-binary-map-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab G)
+
+  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y : type-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+      ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y) ＝
+    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G (add-Metric-Ab G x y)
+  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x y =
+    add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( const-cauchy-approximation-Metric-Ab G x)
+      ( const-cauchy-approximation-Metric-Ab G y) ∙
+    apply-effectiveness-quotient-map'
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( sim-add-const-cauchy-approximation-Metric-Ab G x y)
+```
+
+### Addition in the metric quotient of the Cauchy pseudocompletion is associative
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y z : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+      ( z) ＝
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( x)
+      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y z)
+  associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    triple-induction-set-quotient'
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x y z →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+            ( z))
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( x)
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y z)))
+      ( λ x y z →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+        in
+          equational-reasoning
+            (in-approx-G x +∙G in-approx-G y) +∙G in-approx-G z
+            ＝ in-approx-G (x +~G y) +∙G in-approx-G z
+              by
+                ap-binary
+                  ( _+∙G_)
+                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( x)
+                    ( y))
+                  ( refl)
+            ＝ in-approx-G ((x +~G y) +~G z)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( x +~G y)
+                  ( z)
+            ＝ in-approx-G (x +~G (y +~G z))
+              by
+                apply-effectiveness-quotient-map'
+                  ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab
+                    ( G))
+                  ( sim-associative-add-cauchy-approximation-Metric-Ab G x y z)
+            ＝ in-approx-G x +∙G in-approx-G (y +~G z)
+              by
+                inv
+                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( x)
+                    ( y +~G z))
+            ＝ in-approx-G x +∙G (in-approx-G y +∙G in-approx-G z)
+              by
+                ap-binary
+                  ( _+∙G_)
+                  ( refl)
+                  ( inv
+                    ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                      ( G)
+                      ( y)
+                      ( z))))
+```
+
+### Addition in the metric quotient of the Cauchy pseudocompletion is commutative
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y ＝
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y x
+  commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    double-induction-set-quotient'
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x y →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y x))
+      ( λ x y →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+        in
+          equational-reasoning
+            in-approx-G x +∙G in-approx-G y
+            ＝ in-approx-G (x +~G y)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( x)
+                  ( y)
+            ＝ in-approx-G (y +~G x)
+              by
+                ap
+                  ( in-approx-G)
+                  ( commutative-add-cauchy-approximation-Metric-Ab G x y)
+            ＝ in-approx-G y +∙G in-approx-G x
+              by
+                inv
+                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( y)
+                    ( x)))
+```
+
+### Unit laws of addition in the metric quotient of the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( x) ＝
+    x
+  left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    induction-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+            ( x))
+          ( x))
+      ( λ x →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          0-approx-G = zero-cauchy-approximation-Metric-Ab G
+        in
+          equational-reasoning
+            in-approx-G 0-approx-G +∙G in-approx-G x
+            ＝ in-approx-G (0-approx-G +~G x)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( 0-approx-G)
+                  ( x)
+            ＝ in-approx-G x
+              by
+                apply-effectiveness-quotient-map'
+                  ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab
+                    ( G))
+                  ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab G x))
+
+  right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( x)
+      ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) ＝
+    x
+  right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _ ∙
+    left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x
+```
+
+### Negation in the metric quotient of the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  hom-neg-cauchy-pseudocompletion-Metric-Ab :
+    hom-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+  hom-neg-cauchy-pseudocompletion-Metric-Ab =
+    ( neg-cauchy-approximation-Metric-Ab G ,
+      preserves-sim-neg-cauchy-approximation-Metric-Ab G _ _)
+
+  neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    map-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( hom-neg-cauchy-pseudocompletion-Metric-Ab)
+
+  abstract
+    neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      (x : cauchy-approximation-Metric-Ab G) →
+      neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+          ( G)
+          ( x)) ＝
+      in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( neg-cauchy-approximation-Metric-Ab G x)
+    neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+      coherence-square-map-set-quotient
+        ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+        ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+        ( hom-neg-cauchy-pseudocompletion-Metric-Ab)
+```
+
+### Inverse laws of addition in the metric quotient of the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+      ( x) ＝
+    zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    induction-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+            ( x))
+          ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+      ( λ x →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          neg-∙G = neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          neg-G = neg-cauchy-approximation-Metric-Ab G
+          0-approx-G = zero-cauchy-approximation-Metric-Ab G
+        in
+          equational-reasoning
+            neg-∙G (in-approx-G x) +∙G in-approx-G x
+            ＝ in-approx-G (neg-G x) +∙G in-approx-G x
+              by
+                ap-binary
+                  ( _+∙G_)
+                  ( neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( x))
+                  ( refl)
+            ＝ in-approx-G (neg-G x +~G x)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( neg-G x)
+                  ( x)
+            ＝ in-approx-G 0-approx-G
+              by
+                ap
+                  ( in-approx-G)
+                  ( left-inverse-law-add-cauchy-approximation-Metric-Ab G x))
+
+  right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( x)
+      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x) ＝
+    zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _ ∙
+    left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x
+```
+
+### The metric quotient of the Cauchy pseudocompletion forms an abelian group
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Semigroup (l1 ⊔ l2)
+  semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+
+  is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    is-unital-Semigroup
+      ( semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+  is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+
+  group-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Group (l1 ⊔ l2)
+  group-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+
+  ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Ab (l1 ⊔ l2)
+  ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( group-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+```
diff --git a/src/analysis/metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups.lagda.md
index bf9a4eaf0c3..661fe40eb4c 100644
--- a/src/analysis/metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups.lagda.md
@@ -7,6 +7,7 @@ module analysis.metric-abelian-groups where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
@@ -17,6 +18,7 @@ open import foundation.dependent-pair-types
 open import foundation.dependent-products-propositions
 open import foundation.function-extensionality
 open import foundation.identity-types
+open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -29,8 +31,11 @@ open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-maps-metric-spaces
+open import metric-spaces.monotonic-rational-neighborhood-relations
 open import metric-spaces.pseudometric-spaces
 open import metric-spaces.rational-neighborhood-relations
+open import metric-spaces.reflexive-rational-neighborhood-relations
+open import metric-spaces.triangular-rational-neighborhood-relations
 open import metric-spaces.uniformly-continuous-maps-metric-spaces
 ```
 
@@ -89,32 +94,52 @@ module _
 ```agda
 module _
   {l1 l2 : Level} (MG : Metric-Ab l1 l2)
+  (let ab-MG = ab-Metric-Ab MG)
   where
 
   zero-Metric-Ab : type-Metric-Ab MG
-  zero-Metric-Ab = zero-Ab (ab-Metric-Ab MG)
+  zero-Metric-Ab = zero-Ab ab-MG
 
   add-Metric-Ab : type-Metric-Ab MG → type-Metric-Ab MG → type-Metric-Ab MG
-  add-Metric-Ab = add-Ab (ab-Metric-Ab MG)
+  add-Metric-Ab = add-Ab ab-MG
 
   add-Metric-Ab' : type-Metric-Ab MG → type-Metric-Ab MG → type-Metric-Ab MG
-  add-Metric-Ab' = add-Ab' (ab-Metric-Ab MG)
+  add-Metric-Ab' = add-Ab' ab-MG
 
   ap-add-Metric-Ab :
     {x x' y y' : type-Metric-Ab MG} → x ＝ x' → y ＝ y' →
     add-Metric-Ab x y ＝ add-Metric-Ab x' y'
-  ap-add-Metric-Ab = ap-add-Ab (ab-Metric-Ab MG)
+  ap-add-Metric-Ab = ap-add-Ab ab-MG
 
   neg-Metric-Ab : type-Metric-Ab MG → type-Metric-Ab MG
-  neg-Metric-Ab = neg-Ab (ab-Metric-Ab MG)
+  neg-Metric-Ab = neg-Ab ab-MG
 
   abstract
+    left-unit-law-add-Metric-Ab :
+      (x : type-Metric-Ab MG) → add-Metric-Ab zero-Metric-Ab x ＝ x
+    left-unit-law-add-Metric-Ab = left-unit-law-add-Ab ab-MG
+
+    associative-add-Metric-Ab :
+      (x y z : type-Metric-Ab MG) →
+      add-Metric-Ab (add-Metric-Ab x y) z ＝ add-Metric-Ab x (add-Metric-Ab y z)
+    associative-add-Metric-Ab = associative-add-Ab ab-MG
+
+    left-inverse-law-add-Metric-Ab :
+      (x : type-Metric-Ab MG) →
+      add-Metric-Ab (neg-Metric-Ab x) x ＝ zero-Metric-Ab
+    left-inverse-law-add-Metric-Ab = left-inverse-law-add-Ab ab-MG
+
+    right-inverse-law-add-Metric-Ab :
+      (x : type-Metric-Ab MG) →
+      add-Metric-Ab x (neg-Metric-Ab x) ＝ zero-Metric-Ab
+    right-inverse-law-add-Metric-Ab = right-inverse-law-add-Ab ab-MG
+
     neg-zero-Metric-Ab : neg-Metric-Ab zero-Metric-Ab ＝ zero-Metric-Ab
-    neg-zero-Metric-Ab = neg-zero-Ab (ab-Metric-Ab MG)
+    neg-zero-Metric-Ab = neg-zero-Ab ab-MG
 
     neg-neg-Metric-Ab :
       (x : type-Metric-Ab MG) → neg-Metric-Ab (neg-Metric-Ab x) ＝ x
-    neg-neg-Metric-Ab = neg-neg-Ab (ab-Metric-Ab MG)
+    neg-neg-Metric-Ab = neg-neg-Ab ab-MG
 
   diff-Metric-Ab : type-Metric-Ab MG → type-Metric-Ab MG → type-Metric-Ab MG
   diff-Metric-Ab x y = add-Metric-Ab x (neg-Metric-Ab y)
@@ -122,16 +147,16 @@ module _
   ap-diff-Metric-Ab :
     {x x' y y' : type-Metric-Ab MG} → x ＝ x' → y ＝ y' →
     diff-Metric-Ab x y ＝ diff-Metric-Ab x' y'
-  ap-diff-Metric-Ab = ap-right-subtraction-Ab (ab-Metric-Ab MG)
+  ap-diff-Metric-Ab = ap-right-subtraction-Ab ab-MG
 
   commutative-add-Metric-Ab :
     (x y : type-Metric-Ab MG) → add-Metric-Ab x y ＝ add-Metric-Ab y x
-  commutative-add-Metric-Ab = commutative-add-Ab (ab-Metric-Ab MG)
+  commutative-add-Metric-Ab = commutative-add-Ab ab-MG
 
   is-identity-right-conjugation-Metric-Ab :
     (x y : type-Metric-Ab MG) → add-Metric-Ab x (diff-Metric-Ab y x) ＝ y
   is-identity-right-conjugation-Metric-Ab =
-    is-identity-right-conjugation-Ab (ab-Metric-Ab MG)
+    is-identity-right-conjugation-Ab ab-MG
 ```
 
 ### Metric properties of metric abelian groups
@@ -162,6 +187,21 @@ module _
   neighborhood-Metric-Ab : ℚ⁺ → Relation l2 (type-Metric-Ab MG)
   neighborhood-Metric-Ab = neighborhood-Metric-Space metric-space-Metric-Ab
 
+  refl-neighborhood-Metric-Ab :
+    is-reflexive-Rational-Neighborhood-Relation neighborhood-prop-Metric-Ab
+  refl-neighborhood-Metric-Ab =
+    refl-neighborhood-Metric-Space metric-space-Metric-Ab
+
+  monotonic-neighborhood-Metric-Ab :
+    is-monotonic-Rational-Neighborhood-Relation neighborhood-prop-Metric-Ab
+  monotonic-neighborhood-Metric-Ab =
+    monotonic-neighborhood-Metric-Space metric-space-Metric-Ab
+
+  triangular-neighborhood-Metric-Ab :
+    is-triangular-Rational-Neighborhood-Relation neighborhood-prop-Metric-Ab
+  triangular-neighborhood-Metric-Ab =
+    triangular-neighborhood-Metric-Space metric-space-Metric-Ab
+
   is-isometry-add-Metric-Ab :
     (x : type-Metric-Ab MG) →
     is-isometry-Metric-Space
@@ -253,3 +293,35 @@ module _
       ( metric-space-Metric-Ab G)
       ( modulated-uniformly-continuous-map-add-pair-Metric-Ab)
 ```
+
+### Neighborhoods of sums in metric abelian groups
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (dxx' dyy' : ℚ⁺)
+  (x x' y y' : type-Metric-Ab G)
+  where abstract
+
+  neighborhood-add-Metric-Ab :
+    neighborhood-Metric-Ab G dxx' x x' →
+    neighborhood-Metric-Ab G dyy' y y' →
+    neighborhood-Metric-Ab G
+      ( dxx' +ℚ⁺ dyy')
+      ( add-Metric-Ab G x y)
+      ( add-Metric-Ab G x' y')
+  neighborhood-add-Metric-Ab Nxx' Nyy' =
+    triangular-neighborhood-Metric-Ab G
+      ( add-Metric-Ab G x y)
+      ( add-Metric-Ab G x' y)
+      ( add-Metric-Ab G x' y')
+      ( dxx')
+      ( dyy')
+      ( forward-implication
+        ( is-isometry-add-Metric-Ab G x' dyy' y y')
+        ( Nyy'))
+      ( forward-implication
+        ( is-isometry-add-Metric-Ab' G y dxx' x x')
+        ( Nxx'))
+```
diff --git a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
index 4e2a1657a36..4ca8e619ac0 100644
--- a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
@@ -15,7 +15,9 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.function-types
+open import foundation.identity-types
 open import foundation.propositions
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import order-theory.strict-preorders
@@ -64,6 +66,17 @@ leq-le-ℚ⁺ : {x y : ℚ⁺} → le-ℚ⁺ x y → leq-ℚ⁺ x y
 leq-le-ℚ⁺ {x} {y} = leq-le-ℚ {rational-ℚ⁺ x} {rational-ℚ⁺ y}
 ```
 
+### Concatenation of equality and strict inequality
+
+```agda
+module _
+  {x y z : ℚ⁺}
+  where abstract
+
+  concat-eq-le-ℚ⁺ : x ＝ y → le-ℚ⁺ y z → le-ℚ⁺ x z
+  concat-eq-le-ℚ⁺ = inv-tr (λ w → le-ℚ⁺ w z)
+```
+
 ### The strictly preordered set of positive rational numbers
 
 ```agda
diff --git a/src/foundation/functoriality-set-quotients.lagda.md b/src/foundation/functoriality-set-quotients.lagda.md
index 31b756d2f8c..eaf92db532a 100644
--- a/src/foundation/functoriality-set-quotients.lagda.md
+++ b/src/foundation/functoriality-set-quotients.lagda.md
@@ -41,7 +41,8 @@ open import foundation-core.torsorial-type-families
 
 ## Idea
 
-Set quotients act functorially on types equipped with equivalence relations.
+[Set quotients](foundation.set-quotients.md) act functorially on types equipped
+with [equivalence relations](foundation.equivalence-relations.md).
 
 ## Definition
 
diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index 89f28adc578..5cdbdf07905 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -62,6 +62,9 @@ metric space, `N d₂ x y` [or](foundation.disjunction.md)
 module metric-spaces where
 
 open import metric-spaces.accumulation-points-subsets-located-metric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-short-maps-metric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-uniformly-continuous-maps-metric-spaces public
 open import metric-spaces.action-on-convergent-sequences-modulated-uniformly-continuous-maps-metric-spaces public
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
new file mode 100644
index 00000000000..86ed632d424
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
@@ -0,0 +1,72 @@
+# The action on Cauchy approximations of isometries on metric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.functoriality-set-quotients
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.metric-spaces
+```
+
+</details>
+
+## Idea
+
+The action of [isometries](metric-spaces.isometries-metric-spaces.md) on
+[metric spaces](metric-spaces.metric-spaces.md) preserves
+[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : isometry-Metric-Space X Y)
+  where
+
+  map-isometry-cauchy-approximation-Metric-Space :
+    cauchy-approximation-Metric-Space X →
+    cauchy-approximation-Metric-Space Y
+  map-isometry-cauchy-approximation-Metric-Space =
+    map-short-map-cauchy-approximation-Metric-Space
+      ( X)
+      ( Y)
+      ( short-map-isometry-Metric-Space X Y f)
+```
+
+## Properties
+
+### Isometries preserve similarity in the Cauchy pseudocompletion of a metric space
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : isometry-Metric-Space X Y)
+  where abstract
+
+  preserves-sim-isometry-cauchy-pseudocompletion-Metric-Space :
+    preserves-sim-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space X)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space Y)
+      ( map-isometry-cauchy-approximation-Metric-Space X Y f)
+  preserves-sim-isometry-cauchy-pseudocompletion-Metric-Space {x} {y} =
+    preserves-sim-short-map-cauchy-pseudocompletion-Metric-Space
+      ( X)
+      ( Y)
+      ( short-map-isometry-Metric-Space X Y f)
+      { x}
+      { y}
+```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
new file mode 100644
index 00000000000..bc5271a88d0
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
@@ -0,0 +1,72 @@
+# The action on Cauchy approximations of short maps in metric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.functoriality-set-quotients
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.short-maps-metric-spaces
+```
+
+</details>
+
+## Idea
+
+The action of [short maps](metric-spaces.short-maps-metric-spaces.md) on
+[metric spaces](metric-spaces.metric-spaces.md) preserves
+[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : short-map-Metric-Space X Y)
+  where
+
+  map-short-map-cauchy-approximation-Metric-Space :
+    cauchy-approximation-Metric-Space X →
+    cauchy-approximation-Metric-Space Y
+  map-short-map-cauchy-approximation-Metric-Space =
+    map-short-map-cauchy-approximation-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
+```
+
+## Properties
+
+### Short maps preserve similarity in the Cauchy pseudocompletion of a pseudometric space
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : short-map-Metric-Space X Y)
+  where abstract
+
+  preserves-sim-short-map-cauchy-pseudocompletion-Metric-Space :
+    preserves-sim-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space X)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space Y)
+      ( map-short-map-cauchy-approximation-Metric-Space X Y f)
+  preserves-sim-short-map-cauchy-pseudocompletion-Metric-Space {x} {y} =
+    preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
+      { x}
+      { y}
+```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
new file mode 100644
index 00000000000..6ddcdb67309
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
@@ -0,0 +1,84 @@
+# The action on Cauchy approximations of short maps in pseudometric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.functoriality-set-quotients
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-approximations-pseudometric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
+open import metric-spaces.pseudometric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The action of [short maps](metric-spaces.short-maps-pseudometric-spaces.md) on
+[pseudometric spaces](metric-spaces.pseudometric-spaces.md) preserves
+[Cauchy approximations](metric-spaces.cauchy-approximations-pseudometric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
+  (x@(map-x , is-approx-x) : cauchy-approximation-Pseudometric-Space X)
+  where
+
+  map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space :
+    ℚ⁺ → type-Pseudometric-Space Y
+  map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space
+    ε =
+    map-f (map-x ε)
+
+  abstract
+    is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space :
+      is-cauchy-approximation-Pseudometric-Space
+        ( Y)
+        ( map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space)
+    is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space
+      δ ε =
+      is-short-f (δ +ℚ⁺ ε) (map-x δ) (map-x ε) (is-approx-x δ ε)
+
+  map-short-map-cauchy-approximation-Pseudometric-Space :
+    cauchy-approximation-Pseudometric-Space Y
+  map-short-map-cauchy-approximation-Pseudometric-Space =
+    ( map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space ,
+      is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space)
+```
+
+## Properties
+
+### Short maps preserve similarity in the Cauchy pseudocompletion of a pseudometric space
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
+  where abstract
+
+  preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space :
+    preserves-sim-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space X)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space Y)
+      ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f)
+  preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space
+    {x , is-approx-x} {y , is-approx-y} x~y δ ε θ =
+    is-short-f (ε +ℚ⁺ θ +ℚ⁺ δ) (x ε) (y θ) (x~y δ ε θ)
+```
diff --git a/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md b/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md
index b434d07f104..edb2a44f4f2 100644
--- a/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md
@@ -13,6 +13,7 @@ open import elementary-number-theory.positive-rational-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
+open import foundation.equivalence-relations
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
@@ -253,3 +254,57 @@ module _
     isometry-lim-cauchy-approximation-cauchy-pseudocompletion-Pseudometric-Space
       ( pseudometric-Metric-Space M)
 ```
+
+### The similarity relation in a Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  where
+
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space :
+    equivalence-relation l2 (cauchy-approximation-Metric-Space M)
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space =
+    equivalence-relation-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space M)
+
+  sim-prop-cauchy-pseudocompletion-Metric-Space :
+    Relation-Prop l2 (cauchy-approximation-Metric-Space M)
+  sim-prop-cauchy-pseudocompletion-Metric-Space =
+    sim-prop-Pseudometric-Space (cauchy-pseudocompletion-Metric-Space M)
+
+  sim-cauchy-pseudocompletion-Metric-Space :
+    Relation l2 (cauchy-approximation-Metric-Space M)
+  sim-cauchy-pseudocompletion-Metric-Space =
+    type-Relation-Prop sim-prop-cauchy-pseudocompletion-Metric-Space
+```
+
+### If two constant Cauchy approximations are similar, they have the same constant
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  where abstract
+
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Space :
+    (x y : type-Metric-Space M) →
+    sim-cauchy-pseudocompletion-Metric-Space M
+      ( const-cauchy-approximation-Metric-Space M x)
+      ( const-cauchy-approximation-Metric-Space M y) →
+    x ＝ y
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Space
+    x y cx~cy =
+    eq-sim-Metric-Space
+      ( M)
+      ( x)
+      ( y)
+      ( λ ε →
+        let
+          (ε12 , ε3 , ε12+ε3=ε) = split-ℚ⁺ ε
+          (ε1 , ε2 , ε1+ε2=ε12) = split-ℚ⁺ ε12
+        in
+          tr
+            ( λ d → neighborhood-Metric-Space M d x y)
+            ( ap-add-ℚ⁺ ε1+ε2=ε12 refl ∙ ε12+ε3=ε)
+            ( cx~cy ε3 ε1 ε2))
+```
diff --git a/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md b/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
index 7b9c060eed8..205625ab962 100644
--- a/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
@@ -19,6 +19,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.dependent-products-propositions
+open import foundation.equivalence-relations
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
@@ -279,6 +280,30 @@ module _
       is-saturated-neighborhood-cauchy-pseudocompletion-Pseudometric-Space M)
 ```
 
+### The similarity equivalence relation in the Cauchy pseudocompletion of a pseudometric space
+
+```agda
+module _
+  {l1 l2 : Level} (M : Pseudometric-Space l1 l2)
+  where
+
+  equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space :
+    equivalence-relation l2 (cauchy-approximation-Pseudometric-Space M)
+  equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space =
+    equivalence-relation-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space M)
+
+  sim-prop-cauchy-pseudocompletion-Pseudometric-Space :
+    Relation-Prop l2 (cauchy-approximation-Pseudometric-Space M)
+  sim-prop-cauchy-pseudocompletion-Pseudometric-Space =
+    sim-prop-Pseudometric-Space (cauchy-pseudocompletion-Pseudometric-Space M)
+
+  sim-cauchy-pseudocompletion-Pseudometric-Space :
+    Relation l2 (cauchy-approximation-Pseudometric-Space M)
+  sim-cauchy-pseudocompletion-Pseudometric-Space =
+    sim-Pseudometric-Space (cauchy-pseudocompletion-Pseudometric-Space M)
+```
+
 ### The isometry from a pseudometric space to its Cauchy pseudocompletion
 
 ```agda

From 9ef6bc216e88f373fee7ec9570eb34edecbfca5c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:09:58 -0700
Subject: [PATCH 02/26] Progress

---
 config/codespell-ignore.txt                   |   1 +
 src/analysis.lagda.md                         |   1 +
 ...ocompletion-metric-abelian-groups.lagda.md | 562 +++++++++++++++
 ...roximations-metric-abelian-groups.lagda.md |  74 +-
 ...completions-metric-abelian-groups.lagda.md |  13 +
 ...ocompletion-metric-abelian-groups.lagda.md | 643 +++++-------------
 src/analysis/metric-abelian-groups.lagda.md   |  65 +-
 src/metric-spaces.lagda.md                    |   2 +-
 ...imations-isometries-metric-spaces.lagda.md |  36 +-
 ...ns-isometries-pseudometric-spaces.lagda.md | 102 +++
 ...imations-short-maps-metric-spaces.lagda.md |  72 --
 ...ns-short-maps-pseudometric-spaces.lagda.md |  42 +-
 12 files changed, 1027 insertions(+), 586 deletions(-)
 create mode 100644 src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
 delete mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md

diff --git a/config/codespell-ignore.txt b/config/codespell-ignore.txt
index 79a8f895b86..e060a3a5f41 100644
--- a/config/codespell-ignore.txt
+++ b/config/codespell-ignore.txt
@@ -1,3 +1,4 @@
+Nd
 Tim
 blacklist
 couldn
diff --git a/src/analysis.lagda.md b/src/analysis.lagda.md
index 38b2145e149..6a619f132c2 100644
--- a/src/analysis.lagda.md
+++ b/src/analysis.lagda.md
@@ -3,6 +3,7 @@
 ```agda
 module analysis where
 
+open import analysis.abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups public
 open import analysis.addition-cauchy-approximations-metric-abelian-groups public
 open import analysis.alternation-sequences-metric-abelian-groups public
 open import analysis.cauchy-approximations-metric-abelian-groups public
diff --git a/src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md b/src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..db37e3e37b8
--- /dev/null
+++ b/src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
@@ -0,0 +1,562 @@
+# The abelian group of the metric quotient of Cauchy pseudocompletions of metric abelian groups
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module analysis.abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.addition-cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-functoriality-set-quotients
+open import foundation.binary-relations
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.embeddings
+open import foundation.equivalence-relations
+open import foundation.functoriality-set-quotients
+open import foundation.identity-types
+open import foundation.injective-maps
+open import foundation.propositional-truncations
+open import foundation.set-quotients
+open import foundation.sets
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import group-theory.abelian-groups
+open import group-theory.groups
+open import group-theory.monoids
+open import group-theory.semigroups
+
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.metric-quotients-of-pseudometric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.pseudometric-spaces
+open import metric-spaces.rational-neighborhood-relations
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
+open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The [metric quotient](metric-spaces.metric-quotients-of-pseudometric-spaces.md)
+of the
+[Cauchy pseudocompletion](analysis.cauchy-pseudocompletions-metric-abelian-groups.md)
+of a [metric abelian group](analysis.metric-abelian-groups.md) itself forms a
+metric abelian group.
+
+This construction is precisely analogous to the definition of the Cauchy real
+numbers and their definition of addition.
+
+## Definition
+
+### The metric space of the metric quotient
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Metric-Space (l1 ⊔ l2) (l1 ⊔ l2)
+  metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    metric-quotient-Pseudometric-Space (cauchy-pseudocompletion-Metric-Ab G)
+
+  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Set (l1 ⊔ l2)
+  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    set-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab : UU (l1 ⊔ l2)
+  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    type-Set set-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Rational-Neighborhood-Relation
+      ( l1 ⊔ l2)
+      ( type-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+  neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-prop-Metric-Space
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+
+  neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    ℚ⁺ →
+    Relation (l1 ⊔ l2) type-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+  neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-Metric-Space
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+```
+
+### Addition in the metric quotient
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  binary-hom-add-cauchy-pseudocompletion-Metric-Ab :
+    binary-hom-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+  binary-hom-add-cauchy-pseudocompletion-Metric-Ab =
+    ( add-cauchy-approximation-Metric-Ab G ,
+      λ {x} {x'} {y} {y'} →
+        preserves-sim-add-cauchy-approximation-Metric-Ab G {x} {x'} {y} {y'})
+
+  add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    binary-map-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab)
+```
+
+### The embedding of elements of a metric abelian group in the metric quotient of its Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    cauchy-approximation-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    quotient-map
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+
+  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-Metric-Ab G → type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( const-cauchy-approximation-Metric-Ab G x)
+
+  abstract
+    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      is-injective in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      {x} {y} inx=iny =
+      eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( x)
+        ( y)
+        ( apply-effectiveness-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( inx=iny))
+
+    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      is-emb in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+      is-emb-is-injective
+        ( is-set-type-Metric-Space
+          ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+        is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-Metric-Ab G ↪ type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+
+  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab (zero-Metric-Ab G)
+```
+
+## Properties
+
+### The embedding in the metric quotient of the Cauchy pseudocompletion preserves addition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y : cauchy-approximation-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( x))
+      ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( y)) ＝
+    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( G)
+      ( add-cauchy-approximation-Metric-Ab G x y)
+  add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    compute-binary-map-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab G)
+
+  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y : type-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+      ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y) ＝
+    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G (add-Metric-Ab G x y)
+  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x y =
+    add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( const-cauchy-approximation-Metric-Ab G x)
+      ( const-cauchy-approximation-Metric-Ab G y) ∙
+    apply-effectiveness-quotient-map'
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( sim-add-const-cauchy-approximation-Metric-Ab G x y)
+```
+
+### Addition in the metric quotient of the Cauchy pseudocompletion is associative
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y z : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+      ( z) ＝
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( x)
+      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y z)
+  associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    triple-induction-set-quotient'
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x y z →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+            ( z))
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( x)
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y z)))
+      ( λ x y z →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+        in
+          equational-reasoning
+            (in-approx-G x +∙G in-approx-G y) +∙G in-approx-G z
+            ＝ in-approx-G (x +~G y) +∙G in-approx-G z
+              by
+                ap-binary
+                  ( _+∙G_)
+                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( x)
+                    ( y))
+                  ( refl)
+            ＝ in-approx-G ((x +~G y) +~G z)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( x +~G y)
+                  ( z)
+            ＝ in-approx-G (x +~G (y +~G z))
+              by
+                apply-effectiveness-quotient-map'
+                  ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab
+                    ( G))
+                  ( sim-associative-add-cauchy-approximation-Metric-Ab G x y z)
+            ＝ in-approx-G x +∙G in-approx-G (y +~G z)
+              by
+                inv
+                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( x)
+                    ( y +~G z))
+            ＝ in-approx-G x +∙G (in-approx-G y +∙G in-approx-G z)
+              by
+                ap-binary
+                  ( _+∙G_)
+                  ( refl)
+                  ( inv
+                    ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                      ( G)
+                      ( y)
+                      ( z))))
+```
+
+### Addition in the metric quotient of the Cauchy pseudocompletion is commutative
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x y : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y ＝
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y x
+  commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    double-induction-set-quotient'
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x y →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y x))
+      ( λ x y →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+        in
+          equational-reasoning
+            in-approx-G x +∙G in-approx-G y
+            ＝ in-approx-G (x +~G y)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( x)
+                  ( y)
+            ＝ in-approx-G (y +~G x)
+              by
+                ap
+                  ( in-approx-G)
+                  ( commutative-add-cauchy-approximation-Metric-Ab G x y)
+            ＝ in-approx-G y +∙G in-approx-G x
+              by
+                inv
+                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( y)
+                    ( x)))
+```
+
+### Unit laws of addition in the metric quotient of the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( x) ＝
+    x
+  left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    induction-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+            ( x))
+          ( x))
+      ( λ x →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          0-approx-G = zero-cauchy-approximation-Metric-Ab G
+        in
+          equational-reasoning
+            in-approx-G 0-approx-G +∙G in-approx-G x
+            ＝ in-approx-G (0-approx-G +~G x)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( 0-approx-G)
+                  ( x)
+            ＝ in-approx-G x
+              by
+                apply-effectiveness-quotient-map'
+                  ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab
+                    ( G))
+                  ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab G x))
+
+  right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( x)
+      ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) ＝
+    x
+  right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _ ∙
+    left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x
+```
+
+### Negation in the metric quotient of the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  hom-neg-cauchy-pseudocompletion-Metric-Ab :
+    hom-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+  hom-neg-cauchy-pseudocompletion-Metric-Ab =
+    ( neg-cauchy-approximation-Metric-Ab G ,
+      preserves-sim-neg-cauchy-approximation-Metric-Ab G _ _)
+
+  neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    map-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( hom-neg-cauchy-pseudocompletion-Metric-Ab)
+
+  abstract
+    neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      (x : cauchy-approximation-Metric-Ab G) →
+      neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+          ( G)
+          ( x)) ＝
+      in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+        ( G)
+        ( neg-cauchy-approximation-Metric-Ab G x)
+    neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+      coherence-square-map-set-quotient
+        ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+        ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+        ( hom-neg-cauchy-pseudocompletion-Metric-Ab)
+```
+
+### Inverse laws of addition in the metric quotient of the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+      ( x) ＝
+    zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    induction-set-quotient
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( λ x →
+        Id-Prop
+          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+            ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+            ( x))
+          ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+      ( λ x →
+        let
+          in-approx-G =
+            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          _+~G_ = add-cauchy-approximation-Metric-Ab G
+          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          neg-∙G = neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+          neg-G = neg-cauchy-approximation-Metric-Ab G
+          0-approx-G = zero-cauchy-approximation-Metric-Ab G
+        in
+          equational-reasoning
+            neg-∙G (in-approx-G x) +∙G in-approx-G x
+            ＝ in-approx-G (neg-G x) +∙G in-approx-G x
+              by
+                ap-binary
+                  ( _+∙G_)
+                  ( neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                    ( G)
+                    ( x))
+                  ( refl)
+            ＝ in-approx-G (neg-G x +~G x)
+              by
+                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                  ( G)
+                  ( neg-G x)
+                  ( x)
+            ＝ in-approx-G 0-approx-G
+              by
+                ap
+                  ( in-approx-G)
+                  ( left-inverse-law-add-cauchy-approximation-Metric-Ab G x))
+
+  right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
+    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+      ( x)
+      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x) ＝
+    zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _ ∙
+    left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x
+```
+
+### The metric quotient of the Cauchy pseudocompletion forms an abelian group
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Semigroup (l1 ⊔ l2)
+  semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+
+  is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    is-unital-Semigroup
+      ( semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+  is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+
+  group-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Group (l1 ⊔ l2)
+  group-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+
+  ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Ab (l1 ⊔ l2)
+  ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( group-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+```
diff --git a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
index 340ee0f9d73..15b5c86afe3 100644
--- a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -28,8 +28,10 @@ open import foundation.universe-levels
 open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
 open import metric-spaces.cauchy-approximations-metric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-quotients-of-pseudometric-spaces
 open import metric-spaces.metric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
 open import metric-spaces.similarity-of-elements-pseudometric-spaces
 ```
 
@@ -386,11 +388,69 @@ module _
     sim-cauchy-pseudocompletion-Metric-Ab G
       ( neg-cauchy-approximation-Metric-Ab G x)
       ( neg-cauchy-approximation-Metric-Ab G y)
-  preserves-sim-neg-cauchy-approximation-Metric-Ab x y =
-    preserves-sim-isometry-cauchy-pseudocompletion-Metric-Space
-      ( metric-space-Metric-Ab G)
-      ( metric-space-Metric-Ab G)
-      ( isometry-neg-Metric-Ab G)
-      { x}
-      { y}
+  preserves-sim-neg-cauchy-approximation-Metric-Ab =
+    preserves-sim-map-isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( isometry-cauchy-pseudocompletion-isometry-Metric-Space
+        ( metric-space-Metric-Ab G)
+        ( metric-space-Metric-Ab G)
+        ( isometry-neg-Metric-Ab G))
+```
+
+### Left addition preserves neighborhoods
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  abstract opaque
+    unfolding map-add-cauchy-approximation-Metric-Ab
+
+    preserves-neighborhood-left-add-cauchy-approximation-Metric-Ab :
+      (x : cauchy-approximation-Metric-Ab G) →
+      is-short-map-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( add-cauchy-approximation-Metric-Ab G x)
+    preserves-neighborhood-left-add-cauchy-approximation-Metric-Ab
+      (x , is-approx-x) d (y , is-approx-y) (z , is-approx-z) Ndyz δ ε =
+      let
+        (δ' , 2δ'<δ) = bound-double-le-ℚ⁺ δ
+        (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      in
+        monotonic-neighborhood-Metric-Ab G
+          ( add-Metric-Ab G (x δ') (y δ'))
+          ( add-Metric-Ab G (x ε') (z ε'))
+          ( (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε' +ℚ⁺ d))
+          ( δ +ℚ⁺ ε +ℚ⁺ d)
+          ( concat-eq-le-ℚ⁺
+            { z = δ +ℚ⁺ ε +ℚ⁺ d}
+            ( equational-reasoning
+              (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε' +ℚ⁺ d)
+              ＝ (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε') +ℚ⁺ d
+                by inv (associative-add-ℚ⁺ _ _ _)
+              ＝ (δ' +ℚ⁺ δ') +ℚ⁺ (ε' +ℚ⁺ ε') +ℚ⁺ d
+                by ap-add-ℚ⁺ (interchange-law-add-add-ℚ⁺ _ _ _ _) refl)
+            ( preserves-le-left-add-ℚ _ _ _ (preserves-le-add-ℚ 2δ'<δ 2ε'<ε)))
+          ( neighborhood-add-Metric-Ab G
+            ( δ' +ℚ⁺ ε')
+            ( δ' +ℚ⁺ ε' +ℚ⁺ d)
+            ( x δ')
+            ( x ε')
+            ( y δ')
+            ( z ε')
+            ( is-approx-x δ' ε')
+            ( Ndyz δ' ε'))
+
+  short-map-left-add-cauchy-pseudocompletion-Metric-Ab :
+    (x : cauchy-approximation-Metric-Ab G) →
+    short-map-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+  short-map-left-add-cauchy-pseudocompletion-Metric-Ab x =
+    ( add-cauchy-approximation-Metric-Ab G x ,
+      preserves-neighborhood-left-add-cauchy-approximation-Metric-Ab x)
 ```
diff --git a/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 3a21460a5c9..63b1685f2e2 100644
--- a/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -10,6 +10,8 @@ module analysis.cauchy-pseudocompletions-metric-abelian-groups where
 open import analysis.cauchy-approximations-metric-abelian-groups
 open import analysis.metric-abelian-groups
 
+open import elementary-number-theory.positive-rational-numbers
+
 open import foundation.binary-relations
 open import foundation.equivalence-relations
 open import foundation.identity-types
@@ -17,6 +19,7 @@ open import foundation.universe-levels
 
 open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.pseudometric-spaces
+open import metric-spaces.rational-neighborhood-relations
 open import metric-spaces.similarity-of-elements-pseudometric-spaces
 ```
 
@@ -42,6 +45,16 @@ module _
   cauchy-pseudocompletion-Metric-Ab =
     cauchy-pseudocompletion-Metric-Space (metric-space-Metric-Ab G)
 
+  neighborhood-prop-cauchy-pseudocompletion-Metric-Ab :
+    Rational-Neighborhood-Relation l2 (cauchy-approximation-Metric-Ab G)
+  neighborhood-prop-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-prop-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+
+  neighborhood-cauchy-pseudocompletion-Metric-Ab :
+    ℚ⁺ → Relation l2 (cauchy-approximation-Metric-Ab G)
+  neighborhood-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+
   equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab :
     equivalence-relation l2 (cauchy-approximation-Metric-Ab G)
   equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab =
diff --git a/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
index 12edecbfb38..c8d4b7bca3b 100644
--- a/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
@@ -1,4 +1,4 @@
-# The metric abelian group formed by the metric quotient of Cauchy pseudocompletions of metric abelian groups
+# The metric abelian group of the metric quotient of the Cauchy pseudocompletion of metric abelian groups
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -9,38 +9,31 @@ module analysis.metric-abelian-group-metric-quotient-cauchy-pseudocompletion-met
 <details><summary>Imports</summary>
 
 ```agda
+open import analysis.abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups
 open import analysis.addition-cauchy-approximations-metric-abelian-groups
-open import analysis.cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
 
-open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
-open import foundation.binary-functoriality-set-quotients
+open import foundation.binary-transport
 open import foundation.dependent-pair-types
-open import foundation.embeddings
-open import foundation.equivalence-relations
-open import foundation.functoriality-set-quotients
 open import foundation.identity-types
-open import foundation.injective-maps
+open import foundation.propositional-truncations
 open import foundation.set-quotients
-open import foundation.sets
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.abelian-groups
-open import group-theory.groups
-open import group-theory.monoids
-open import group-theory.semigroups
 
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.metric-quotients-of-pseudometric-spaces
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
-open import metric-spaces.pseudometric-spaces
-open import metric-spaces.similarity-of-elements-pseudometric-spaces
+open import metric-spaces.short-maps-metric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
+open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 ```
 
 </details>
@@ -50,349 +43,12 @@ open import metric-spaces.similarity-of-elements-pseudometric-spaces
 The [metric quotient](metric-spaces.metric-quotients-of-pseudometric-spaces.md)
 of the
 [Cauchy pseudocompletion](analysis.cauchy-pseudocompletions-metric-abelian-groups.md)
-of a [metric abelian group](analysis.metric-abelian-groups.md) itself forms a
+of a [metric abelian group](analysis.metric-abelian-groups.md) is itself a
 metric abelian group.
 
-This construction is precisely analogous to the definition of the Cauchy real
-numbers and their definition of addition.
+## Proof
 
-## Definition
-
-### The metric space of the metric quotient
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    Metric-Space (l1 ⊔ l2) (l1 ⊔ l2)
-  metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    metric-quotient-Pseudometric-Space (cauchy-pseudocompletion-Metric-Ab G)
-
-  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Set (l1 ⊔ l2)
-  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    set-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
-
-  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab : UU (l1 ⊔ l2)
-  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    type-Set set-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-```
-
-### Addition in the metric quotient
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  binary-hom-add-cauchy-pseudocompletion-Metric-Ab :
-    binary-hom-equivalence-relation
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-  binary-hom-add-cauchy-pseudocompletion-Metric-Ab =
-    ( add-cauchy-approximation-Metric-Ab G ,
-      λ {x} {x'} {y} {y'} →
-        preserves-sim-add-cauchy-approximation-Metric-Ab G {x} {x'} {y} {y'})
-
-  add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    binary-map-set-quotient
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab)
-```
-
-### The embedding of elements of a metric abelian group in the metric quotient of its Cauchy pseudocompletion
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    cauchy-approximation-Metric-Ab G →
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    quotient-map
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-
-  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-Metric-Ab G → type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
-    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-      ( const-cauchy-approximation-Metric-Ab G x)
-
-  abstract
-    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-      is-injective in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-      {x} {y} inx=iny =
-      eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab
-        ( G)
-        ( x)
-        ( y)
-        ( apply-effectiveness-quotient-map
-          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-          ( inx=iny))
-
-    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-      is-emb in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-      is-emb-is-injective
-        ( is-set-type-Metric-Space
-          ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
-        is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-
-  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-Metric-Ab G ↪ type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
-      is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-
-  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab (zero-Metric-Ab G)
-```
-
-## Properties
-
-### The embedding in the metric quotient of the Cauchy pseudocompletion preserves addition
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where abstract
-
-  add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x y : cauchy-approximation-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-        ( G)
-        ( x))
-      ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-        ( G)
-        ( y)) ＝
-    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-      ( G)
-      ( add-cauchy-approximation-Metric-Ab G x y)
-  add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    compute-binary-map-set-quotient
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab G)
-
-  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x y : type-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
-      ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y) ＝
-    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G (add-Metric-Ab G x y)
-  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x y =
-    add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-      ( const-cauchy-approximation-Metric-Ab G x)
-      ( const-cauchy-approximation-Metric-Ab G y) ∙
-    apply-effectiveness-quotient-map'
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( sim-add-const-cauchy-approximation-Metric-Ab G x y)
-```
-
-### Addition in the metric quotient of the Cauchy pseudocompletion is associative
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where abstract
-
-  associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x y z : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
-      ( z) ＝
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( x)
-      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y z)
-  associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    triple-induction-set-quotient'
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( λ x y z →
-        Id-Prop
-          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
-            ( z))
-          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-            ( x)
-            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y z)))
-      ( λ x y z →
-        let
-          in-approx-G =
-            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          _+~G_ = add-cauchy-approximation-Metric-Ab G
-          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-        in
-          equational-reasoning
-            (in-approx-G x +∙G in-approx-G y) +∙G in-approx-G z
-            ＝ in-approx-G (x +~G y) +∙G in-approx-G z
-              by
-                ap-binary
-                  ( _+∙G_)
-                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                    ( G)
-                    ( x)
-                    ( y))
-                  ( refl)
-            ＝ in-approx-G ((x +~G y) +~G z)
-              by
-                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                  ( G)
-                  ( x +~G y)
-                  ( z)
-            ＝ in-approx-G (x +~G (y +~G z))
-              by
-                apply-effectiveness-quotient-map'
-                  ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab
-                    ( G))
-                  ( sim-associative-add-cauchy-approximation-Metric-Ab G x y z)
-            ＝ in-approx-G x +∙G in-approx-G (y +~G z)
-              by
-                inv
-                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                    ( G)
-                    ( x)
-                    ( y +~G z))
-            ＝ in-approx-G x +∙G (in-approx-G y +∙G in-approx-G z)
-              by
-                ap-binary
-                  ( _+∙G_)
-                  ( refl)
-                  ( inv
-                    ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                      ( G)
-                      ( y)
-                      ( z))))
-```
-
-### Addition in the metric quotient of the Cauchy pseudocompletion is commutative
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where abstract
-
-  commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x y : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y ＝
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y x
-  commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    double-induction-set-quotient'
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( λ x y →
-        Id-Prop
-          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
-          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y x))
-      ( λ x y →
-        let
-          in-approx-G =
-            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          _+~G_ = add-cauchy-approximation-Metric-Ab G
-          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-        in
-          equational-reasoning
-            in-approx-G x +∙G in-approx-G y
-            ＝ in-approx-G (x +~G y)
-              by
-                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                  ( G)
-                  ( x)
-                  ( y)
-            ＝ in-approx-G (y +~G x)
-              by
-                ap
-                  ( in-approx-G)
-                  ( commutative-add-cauchy-approximation-Metric-Ab G x y)
-            ＝ in-approx-G y +∙G in-approx-G x
-              by
-                inv
-                  ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                    ( G)
-                    ( y)
-                    ( x)))
-```
-
-### Unit laws of addition in the metric quotient of the Cauchy pseudocompletion
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where abstract
-
-  left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-      ( x) ＝
-    x
-  left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    induction-set-quotient
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( λ x →
-        Id-Prop
-          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-            ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-            ( x))
-          ( x))
-      ( λ x →
-        let
-          in-approx-G =
-            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          _+~G_ = add-cauchy-approximation-Metric-Ab G
-          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          0-approx-G = zero-cauchy-approximation-Metric-Ab G
-        in
-          equational-reasoning
-            in-approx-G 0-approx-G +∙G in-approx-G x
-            ＝ in-approx-G (0-approx-G +~G x)
-              by
-                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                  ( G)
-                  ( 0-approx-G)
-                  ( x)
-            ＝ in-approx-G x
-              by
-                apply-effectiveness-quotient-map'
-                  ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab
-                    ( G))
-                  ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab G x))
-
-  right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( x)
-      ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) ＝
-    x
-  right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
-    commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _ ∙
-    left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x
-```
-
-### Negation in the metric quotient of the Cauchy pseudocompletion
+### Negation is a short map
 
 ```agda
 module _
@@ -400,41 +56,75 @@ module _
   (G : Metric-Ab l1 l2)
   where abstract
 
-  hom-neg-cauchy-pseudocompletion-Metric-Ab :
-    hom-equivalence-relation
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-  hom-neg-cauchy-pseudocompletion-Metric-Ab =
-    ( neg-cauchy-approximation-Metric-Ab G ,
-      preserves-sim-neg-cauchy-approximation-Metric-Ab G _ _)
-
-  neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    map-set-quotient
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( hom-neg-cauchy-pseudocompletion-Metric-Ab)
-
-  abstract
-    neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-      (x : cauchy-approximation-Metric-Ab G) →
-      neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-        ( in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-          ( G)
-          ( x)) ＝
-      in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-        ( G)
-        ( neg-cauchy-approximation-Metric-Ab G x)
-    neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-      coherence-square-map-set-quotient
-        ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-        ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-        ( hom-neg-cauchy-pseudocompletion-Metric-Ab)
+  is-short-map-neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    is-short-map-Metric-Space
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+  is-short-map-neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    d x y Ndxy =
+    let
+      open
+        do-syntax-trunc-Prop
+          ( neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+            ( G)
+            ( d)
+            ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+            ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y))
+    in do
+      (x' , ux'=x) ←
+        is-surjective-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( x)
+      (y' , uy'=y) ←
+        is-surjective-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( y)
+      binary-tr
+        ( neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab G d)
+        ( ( inv
+            ( neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+              ( G)
+              ( x'))) ∙
+          ( ap (neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) ux'=x))
+        ( ( inv
+            ( neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+              ( G)
+              ( y'))) ∙
+          ( ap (neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) uy'=y))
+        ( preserves-neighborhoods-map-isometry-Pseudometric-Space
+          ( cauchy-pseudocompletion-Metric-Ab G)
+          ( pseudometric-Metric-Space
+            ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+          ( comp-isometry-Pseudometric-Space
+            ( cauchy-pseudocompletion-Metric-Ab G)
+            ( cauchy-pseudocompletion-Metric-Ab G)
+            ( pseudometric-Metric-Space
+              ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+            ( isometry-unit-metric-quotient-Pseudometric-Space
+              ( cauchy-pseudocompletion-Metric-Ab G))
+            ( isometry-cauchy-pseudocompletion-isometry-Metric-Space
+              ( metric-space-Metric-Ab G)
+              ( metric-space-Metric-Ab G)
+              ( isometry-neg-Metric-Ab G)))
+          ( d)
+          ( x')
+          ( y')
+          ( reflects-neighborhoods-map-unit-metric-quotient-Pseudometric-Space
+            ( cauchy-pseudocompletion-Metric-Ab G)
+            ( d)
+            ( x')
+            ( y')
+            ( binary-tr
+              ( neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                ( G)
+                ( d))
+              ( inv ux'=x)
+              ( inv uy'=y)
+              ( Ndxy))))
 ```
 
-### Inverse laws of addition in the metric quotient of the Cauchy pseudocompletion
+### Left addition is a short map on the metric quotient of the Cauchy pseudocompletion of a metric abelian group
 
 ```agda
 module _
@@ -442,66 +132,85 @@ module _
   (G : Metric-Ab l1 l2)
   where abstract
 
-  left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
-      ( x) ＝
-    zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    induction-set-quotient
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-      ( λ x →
-        Id-Prop
-          ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-          ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-            ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
-            ( x))
-          ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
-      ( λ x →
-        let
-          in-approx-G =
-            in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          _+~G_ = add-cauchy-approximation-Metric-Ab G
-          _+∙G_ = add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          neg-∙G = neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-          neg-G = neg-cauchy-approximation-Metric-Ab G
-          0-approx-G = zero-cauchy-approximation-Metric-Ab G
-        in
-          equational-reasoning
-            neg-∙G (in-approx-G x) +∙G in-approx-G x
-            ＝ in-approx-G (neg-G x) +∙G in-approx-G x
-              by
-                ap-binary
-                  ( _+∙G_)
-                  ( neg-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                    ( G)
-                    ( x))
-                  ( refl)
-            ＝ in-approx-G (neg-G x +~G x)
-              by
-                add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-                  ( G)
-                  ( neg-G x)
-                  ( x)
-            ＝ in-approx-G 0-approx-G
-              by
-                ap
-                  ( in-approx-G)
-                  ( left-inverse-law-add-cauchy-approximation-Metric-Ab G x))
-
-  right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+  is-short-map-left-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
     (x : type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G) →
-    add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-      ( x)
-      ( neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x) ＝
-    zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
-    commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _ ∙
-    left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab x
+    is-short-map-Metric-Space
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
+  is-short-map-left-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    x d y z Ndyz =
+    let
+      open
+        do-syntax-trunc-Prop
+          ( neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+            ( G)
+            ( d)
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x y)
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x z))
+    in do
+      (x' , ux'=x) ←
+        is-surjective-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( x)
+      (y' , uy'=y) ←
+        is-surjective-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( y)
+      (z' , uz'=z) ←
+        is-surjective-quotient-map
+          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+          ( z)
+      binary-tr
+        ( neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab G d)
+        ( ( inv
+            ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+              ( G)
+              ( x')
+              ( y'))) ∙
+          ( ap-binary
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+            ( ux'=x)
+            ( uy'=y)))
+        ( ( inv
+            ( add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+              ( G)
+              ( x')
+              ( z'))) ∙
+          ( ap-binary
+            ( add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+            ( ux'=x)
+            ( uz'=z)))
+        ( is-short-map-short-map-Pseudometric-Space
+          ( cauchy-pseudocompletion-Metric-Ab G)
+          ( pseudometric-Metric-Space
+            ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+          ( comp-short-map-Pseudometric-Space
+            ( cauchy-pseudocompletion-Metric-Ab G)
+            ( cauchy-pseudocompletion-Metric-Ab G)
+            ( pseudometric-Metric-Space
+              ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+            ( short-map-unit-metric-quotient-Pseudometric-Space
+              ( cauchy-pseudocompletion-Metric-Ab G))
+            ( short-map-left-add-cauchy-pseudocompletion-Metric-Ab G x'))
+          ( d)
+          ( y')
+          ( z')
+          ( reflects-neighborhoods-map-unit-metric-quotient-Pseudometric-Space
+            ( cauchy-pseudocompletion-Metric-Ab G)
+            ( d)
+            ( y')
+            ( z')
+            ( binary-tr
+              ( neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+                ( G)
+                ( d))
+              ( inv uy'=y)
+              ( inv uz'=z)
+              ( Ndyz))))
 ```
 
-### The metric quotient of the Cauchy pseudocompletion forms an abelian group
+## Definition
 
 ```agda
 module _
@@ -509,32 +218,14 @@ module _
   (G : Metric-Ab l1 l2)
   where
 
-  semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    Semigroup (l1 ⊔ l2)
-  semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    ( set-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
-      add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
-      associative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-
-  is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    is-unital-Semigroup
-      ( semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-  is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    ( zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
-      left-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
-      right-unit-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-
-  group-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    Group (l1 ⊔ l2)
-  group-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    ( semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
-      is-unital-semigroup-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
-      neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
-      left-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
-      right-inverse-law-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-
-  ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Ab (l1 ⊔ l2)
-  ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    ( group-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
-      commutative-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+  metric-ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Metric-Ab (l1 ⊔ l2) (l1 ⊔ l2)
+  metric-ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      pseudometric-structure-Metric-Space
+        ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G) ,
+      is-extensional-pseudometric-Metric-Space
+        ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G) ,
+      is-short-map-neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      is-short-map-left-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
 ```
diff --git a/src/analysis/metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups.lagda.md
index 661fe40eb4c..57eaaa6771c 100644
--- a/src/analysis/metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups.lagda.md
@@ -12,6 +12,7 @@ open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.binary-relations
+open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
@@ -28,13 +29,14 @@ open import group-theory.abelian-groups
 open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.extensionality-pseudometric-spaces
 open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-maps-metric-spaces
 open import metric-spaces.monotonic-rational-neighborhood-relations
 open import metric-spaces.pseudometric-spaces
 open import metric-spaces.rational-neighborhood-relations
 open import metric-spaces.reflexive-rational-neighborhood-relations
+open import metric-spaces.short-maps-metric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
 open import metric-spaces.triangular-rational-neighborhood-relations
 open import metric-spaces.uniformly-continuous-maps-metric-spaces
 ```
@@ -46,8 +48,9 @@ open import metric-spaces.uniformly-continuous-maps-metric-spaces
 A {{#concept "metric abelian group" Agda=Metric-Ab}} is an
 [abelian group](group-theory.abelian-groups.md) endowed with the structure of a
 [metric space](metric-spaces.metric-spaces.md) such that the addition operation
-and negation operation are
-[isometries](metric-spaces.isometries-metric-spaces.md).
+and negation operation are [short](metric-spaces.short-maps-metric-spaces.md)
+(which, together with the group operations, implies they are
+[isometries](metric-spaces.isometries-metric-spaces.md)).
 
 ## Definition
 
@@ -60,10 +63,10 @@ is-metric-ab-prop-Ab-Pseudometric-Structure G M =
     MS = (type-Ab G , M)
   in
     is-extensional-prop-Pseudometric-Space MS ∧
-    is-isometry-prop-Pseudometric-Space MS MS (neg-Ab G) ∧
+    is-short-map-prop-Pseudometric-Space MS MS (neg-Ab G) ∧
     Π-Prop
       ( type-Ab G)
-      ( λ x → is-isometry-prop-Pseudometric-Space MS MS (add-Ab G x))
+      ( λ x → is-short-map-prop-Pseudometric-Space MS MS (add-Ab G x))
 
 is-metric-ab-Ab-Pseudometric-Structure :
   {l1 l2 : Level} (G : Ab l1) (M : Pseudometric-Structure l2 (type-Ab G)) →
@@ -202,13 +205,40 @@ module _
   triangular-neighborhood-Metric-Ab =
     triangular-neighborhood-Metric-Space metric-space-Metric-Ab
 
+  is-short-map-add-Metric-Ab :
+    (x : type-Metric-Ab MG) →
+    is-short-map-Metric-Space
+      ( metric-space-Metric-Ab)
+      ( metric-space-Metric-Ab)
+      ( add-Metric-Ab MG x)
+  is-short-map-add-Metric-Ab = pr2 (pr2 (pr2 (pr2 MG)))
+
+  abstract
+    reflects-neighborhoods-left-add-Metric-Ab :
+      (x : type-Metric-Ab MG)
+      (d : ℚ⁺)
+      (y z : type-Metric-Ab MG) →
+      neighborhood-Metric-Ab
+        ( d)
+        ( add-Metric-Ab MG x y)
+        ( add-Metric-Ab MG x z) →
+      neighborhood-Metric-Ab d y z
+    reflects-neighborhoods-left-add-Metric-Ab x d y z Nd⟨x+y⟩⟨x+z⟩ =
+      binary-tr
+        ( neighborhood-Metric-Ab d)
+        ( is-retraction-left-subtraction-Ab (ab-Metric-Ab MG) x y)
+        ( is-retraction-left-subtraction-Ab (ab-Metric-Ab MG) x z)
+        ( is-short-map-add-Metric-Ab (neg-Metric-Ab MG x) d _ _ Nd⟨x+y⟩⟨x+z⟩)
+
   is-isometry-add-Metric-Ab :
     (x : type-Metric-Ab MG) →
     is-isometry-Metric-Space
       ( metric-space-Metric-Ab)
       ( metric-space-Metric-Ab)
       ( add-Metric-Ab MG x)
-  is-isometry-add-Metric-Ab = pr2 (pr2 (pr2 (pr2 MG)))
+  is-isometry-add-Metric-Ab x d y z =
+    ( is-short-map-add-Metric-Ab x d y z ,
+      reflects-neighborhoods-left-add-Metric-Ab x d y z)
 
   isometry-add-Metric-Ab :
     (x : type-Metric-Ab MG) →
@@ -240,12 +270,33 @@ module _
   isometry-add-Metric-Ab' x =
     ( add-Metric-Ab' MG x , is-isometry-add-Metric-Ab' x)
 
+  is-short-map-neg-Metric-Ab :
+    is-short-map-Metric-Space
+      ( metric-space-Metric-Ab)
+      ( metric-space-Metric-Ab)
+      ( neg-Metric-Ab MG)
+  is-short-map-neg-Metric-Ab = pr1 (pr2 (pr2 (pr2 MG)))
+
+  abstract
+    reflects-neighborhoods-neg-Metric-Ab :
+      (d : ℚ⁺) (x y : type-Metric-Ab MG) →
+      neighborhood-Metric-Ab d (neg-Metric-Ab MG x) (neg-Metric-Ab MG y) →
+      neighborhood-Metric-Ab d x y
+    reflects-neighborhoods-neg-Metric-Ab d x y Nd⟨-x⟩⟨-y⟩ =
+      binary-tr
+        ( neighborhood-Metric-Ab d)
+        ( neg-neg-Metric-Ab MG x)
+        ( neg-neg-Metric-Ab MG y)
+        ( is-short-map-neg-Metric-Ab d _ _ Nd⟨-x⟩⟨-y⟩)
+
   is-isometry-neg-Metric-Ab :
     is-isometry-Metric-Space
       ( metric-space-Metric-Ab)
       ( metric-space-Metric-Ab)
       ( neg-Metric-Ab MG)
-  is-isometry-neg-Metric-Ab = pr1 (pr2 (pr2 (pr2 MG)))
+  is-isometry-neg-Metric-Ab d x y =
+    ( is-short-map-neg-Metric-Ab d x y ,
+      reflects-neighborhoods-neg-Metric-Ab d x y)
 
   isometry-neg-Metric-Ab :
     isometry-Metric-Space
diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index 5cdbdf07905..2e432f509dd 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -63,7 +63,7 @@ module metric-spaces where
 
 open import metric-spaces.accumulation-points-subsets-located-metric-spaces public
 open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces public
-open import metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces public
 open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-short-maps-metric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-uniformly-continuous-maps-metric-spaces public
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
index 86ed632d424..8f6e86c0afd 100644
--- a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
@@ -10,10 +10,11 @@ module metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces wh
 open import foundation.functoriality-set-quotients
 open import foundation.universe-levels
 
-open import metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces
+open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces
 open import metric-spaces.cauchy-approximations-metric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
 ```
 
@@ -39,15 +40,15 @@ module _
     cauchy-approximation-Metric-Space X →
     cauchy-approximation-Metric-Space Y
   map-isometry-cauchy-approximation-Metric-Space =
-    map-short-map-cauchy-approximation-Metric-Space
-      ( X)
-      ( Y)
-      ( short-map-isometry-Metric-Space X Y f)
+    map-isometry-cauchy-approximation-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
 ```
 
 ## Properties
 
-### Isometries preserve similarity in the Cauchy pseudocompletion of a metric space
+### An isometry on metric spaces induces an isometry on the Cauchy pseudocompletions of the metric spaces
 
 ```agda
 module _
@@ -55,18 +56,15 @@ module _
   (X : Metric-Space l1 l2)
   (Y : Metric-Space l3 l4)
   (f : isometry-Metric-Space X Y)
-  where abstract
+  where
 
-  preserves-sim-isometry-cauchy-pseudocompletion-Metric-Space :
-    preserves-sim-equivalence-relation
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space X)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space Y)
-      ( map-isometry-cauchy-approximation-Metric-Space X Y f)
-  preserves-sim-isometry-cauchy-pseudocompletion-Metric-Space {x} {y} =
-    preserves-sim-short-map-cauchy-pseudocompletion-Metric-Space
-      ( X)
-      ( Y)
-      ( short-map-isometry-Metric-Space X Y f)
-      { x}
-      { y}
+  isometry-cauchy-pseudocompletion-isometry-Metric-Space :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space X)
+      ( cauchy-pseudocompletion-Metric-Space Y)
+  isometry-cauchy-pseudocompletion-isometry-Metric-Space =
+    isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
 ```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
new file mode 100644
index 00000000000..16d2b0789ef
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
@@ -0,0 +1,102 @@
+# The action on Cauchy approximations of isometries in pseudometric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
+open import metric-spaces.cauchy-approximations-pseudometric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
+open import metric-spaces.pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+[Isometries](metric-spaces.isometries-pseudometric-spaces.md) on
+[pseudometric spaces](metric-spaces.pseudometric-spaces.md) induce isometries on
+the
+[Cauchy pseudocompletion](metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces.md)
+of the pseudometric spaces.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f : isometry-Pseudometric-Space X Y)
+  where
+
+  map-isometry-cauchy-approximation-Pseudometric-Space :
+    cauchy-approximation-Pseudometric-Space X →
+    cauchy-approximation-Pseudometric-Space Y
+  map-isometry-cauchy-approximation-Pseudometric-Space =
+    map-short-map-cauchy-approximation-Pseudometric-Space
+      ( X)
+      ( Y)
+      ( short-map-isometry-Pseudometric-Space X Y f)
+```
+
+## Properties
+
+### Mapping an isometry on Cauchy approximations in a pseudometric space is an isometry in the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f : isometry-Pseudometric-Space X Y)
+  where
+
+  abstract
+    is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space :
+      is-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Pseudometric-Space X)
+        ( cauchy-pseudocompletion-Pseudometric-Space Y)
+        ( map-isometry-cauchy-approximation-Pseudometric-Space X Y f)
+    pr1
+      ( is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+        d (x , is-approx-x) (y , is-approx-y))
+      Ndxy δ ε =
+      preserves-neighborhoods-map-isometry-Pseudometric-Space
+        ( X)
+        ( Y)
+        ( f)
+        ( δ +ℚ⁺ ε +ℚ⁺ d)
+        ( x δ)
+        ( y ε)
+        ( Ndxy δ ε)
+    pr2
+      ( is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+        d (x , is-approx-x) (y , is-approx-y))
+      Ndfxfy δ ε =
+      reflects-neighborhoods-map-isometry-Pseudometric-Space
+        ( X)
+        ( Y)
+        ( f)
+        ( δ +ℚ⁺ ε +ℚ⁺ d)
+        ( x δ)
+        ( y ε)
+        ( Ndfxfy δ ε)
+
+  isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space X)
+      ( cauchy-pseudocompletion-Pseudometric-Space Y)
+  isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space =
+    ( map-isometry-cauchy-approximation-Pseudometric-Space X Y f ,
+      is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space)
+```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
deleted file mode 100644
index bc5271a88d0..00000000000
--- a/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
+++ /dev/null
@@ -1,72 +0,0 @@
-# The action on Cauchy approximations of short maps in metric spaces
-
-```agda
-module metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import foundation.functoriality-set-quotients
-open import foundation.universe-levels
-
-open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.metric-spaces
-open import metric-spaces.short-maps-metric-spaces
-```
-
-</details>
-
-## Idea
-
-The action of [short maps](metric-spaces.short-maps-metric-spaces.md) on
-[metric spaces](metric-spaces.metric-spaces.md) preserves
-[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md).
-
-## Definition
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Metric-Space l1 l2)
-  (Y : Metric-Space l3 l4)
-  (f : short-map-Metric-Space X Y)
-  where
-
-  map-short-map-cauchy-approximation-Metric-Space :
-    cauchy-approximation-Metric-Space X →
-    cauchy-approximation-Metric-Space Y
-  map-short-map-cauchy-approximation-Metric-Space =
-    map-short-map-cauchy-approximation-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( pseudometric-Metric-Space Y)
-      ( f)
-```
-
-## Properties
-
-### Short maps preserve similarity in the Cauchy pseudocompletion of a pseudometric space
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Metric-Space l1 l2)
-  (Y : Metric-Space l3 l4)
-  (f : short-map-Metric-Space X Y)
-  where abstract
-
-  preserves-sim-short-map-cauchy-pseudocompletion-Metric-Space :
-    preserves-sim-equivalence-relation
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space X)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space Y)
-      ( map-short-map-cauchy-approximation-Metric-Space X Y f)
-  preserves-sim-short-map-cauchy-pseudocompletion-Metric-Space {x} {y} =
-    preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( pseudometric-Metric-Space Y)
-      ( f)
-      { x}
-      { y}
-```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
index 6ddcdb67309..368391bef67 100644
--- a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
@@ -18,6 +18,7 @@ open import metric-spaces.cauchy-approximations-pseudometric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
 open import metric-spaces.pseudometric-spaces
 open import metric-spaces.short-maps-pseudometric-spaces
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
 ```
 
 </details>
@@ -63,7 +64,7 @@ module _
 
 ## Properties
 
-### Short maps preserve similarity in the Cauchy pseudocompletion of a pseudometric space
+### Short maps preserve neighborhoods in the Cauchy pseudocompletion of a pseudometric space
 
 ```agda
 module _
@@ -71,6 +72,35 @@ module _
   (X : Pseudometric-Space l1 l2)
   (Y : Pseudometric-Space l3 l4)
   (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
+  where
+
+  abstract
+    is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space :
+      is-short-map-Pseudometric-Space
+        ( cauchy-pseudocompletion-Pseudometric-Space X)
+        ( cauchy-pseudocompletion-Pseudometric-Space Y)
+        ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f)
+    is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space
+      d (x , _) (y , _) Ndxy δ ε =
+      is-short-f (δ +ℚ⁺ ε +ℚ⁺ d) (x δ) (y ε) (Ndxy δ ε)
+
+  short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space :
+    short-map-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space X)
+      ( cauchy-pseudocompletion-Pseudometric-Space Y)
+  short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space =
+    ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f ,
+      is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space)
+```
+
+### Short maps preserve similarity in the Cauchy pseudocompletion of a pseudometric space
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f : short-map-Pseudometric-Space X Y)
   where abstract
 
   preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space :
@@ -78,7 +108,11 @@ module _
       ( equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space X)
       ( equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space Y)
       ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f)
-  preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space
-    {x , is-approx-x} {y , is-approx-y} x~y δ ε θ =
-    is-short-f (ε +ℚ⁺ θ +ℚ⁺ δ) (x ε) (y θ) (x~y δ ε θ)
+  preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space {x} {y} =
+    preserves-sim-map-short-map-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space X)
+      ( cauchy-pseudocompletion-Pseudometric-Space Y)
+      ( short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space X Y f)
+      ( x)
+      ( y)
 ```

From 626caa77294ec4596d95ecd1c1fc042805a3c563 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:14:20 -0700
Subject: [PATCH 03/26] Progress

---
 src/metric-spaces.lagda.md                    |   3 +
 ...imations-isometries-metric-spaces.lagda.md |  70 ++++++++++++
 ...ns-isometries-pseudometric-spaces.lagda.md | 102 ++++++++++++++++++
 ...ns-short-maps-pseudometric-spaces.lagda.md |  92 ++++++++++++++++
 ...seudocompletions-of-metric-spaces.lagda.md |  55 ++++++++++
 ...ompletions-of-pseudometric-spaces.lagda.md |  25 +++++
 6 files changed, 347 insertions(+)
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md

diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index 89f28adc578..2e432f509dd 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -62,6 +62,9 @@ metric space, `N d₂ x y` [or](foundation.disjunction.md)
 module metric-spaces where
 
 open import metric-spaces.accumulation-points-subsets-located-metric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-short-maps-metric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-uniformly-continuous-maps-metric-spaces public
 open import metric-spaces.action-on-convergent-sequences-modulated-uniformly-continuous-maps-metric-spaces public
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
new file mode 100644
index 00000000000..8f6e86c0afd
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
@@ -0,0 +1,70 @@
+# The action on Cauchy approximations of isometries on metric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.functoriality-set-quotients
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
+open import metric-spaces.metric-spaces
+```
+
+</details>
+
+## Idea
+
+The action of [isometries](metric-spaces.isometries-metric-spaces.md) on
+[metric spaces](metric-spaces.metric-spaces.md) preserves
+[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : isometry-Metric-Space X Y)
+  where
+
+  map-isometry-cauchy-approximation-Metric-Space :
+    cauchy-approximation-Metric-Space X →
+    cauchy-approximation-Metric-Space Y
+  map-isometry-cauchy-approximation-Metric-Space =
+    map-isometry-cauchy-approximation-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
+```
+
+## Properties
+
+### An isometry on metric spaces induces an isometry on the Cauchy pseudocompletions of the metric spaces
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : isometry-Metric-Space X Y)
+  where
+
+  isometry-cauchy-pseudocompletion-isometry-Metric-Space :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space X)
+      ( cauchy-pseudocompletion-Metric-Space Y)
+  isometry-cauchy-pseudocompletion-isometry-Metric-Space =
+    isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
+```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
new file mode 100644
index 00000000000..16d2b0789ef
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
@@ -0,0 +1,102 @@
+# The action on Cauchy approximations of isometries in pseudometric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
+open import metric-spaces.cauchy-approximations-pseudometric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
+open import metric-spaces.pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+[Isometries](metric-spaces.isometries-pseudometric-spaces.md) on
+[pseudometric spaces](metric-spaces.pseudometric-spaces.md) induce isometries on
+the
+[Cauchy pseudocompletion](metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces.md)
+of the pseudometric spaces.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f : isometry-Pseudometric-Space X Y)
+  where
+
+  map-isometry-cauchy-approximation-Pseudometric-Space :
+    cauchy-approximation-Pseudometric-Space X →
+    cauchy-approximation-Pseudometric-Space Y
+  map-isometry-cauchy-approximation-Pseudometric-Space =
+    map-short-map-cauchy-approximation-Pseudometric-Space
+      ( X)
+      ( Y)
+      ( short-map-isometry-Pseudometric-Space X Y f)
+```
+
+## Properties
+
+### Mapping an isometry on Cauchy approximations in a pseudometric space is an isometry in the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f : isometry-Pseudometric-Space X Y)
+  where
+
+  abstract
+    is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space :
+      is-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Pseudometric-Space X)
+        ( cauchy-pseudocompletion-Pseudometric-Space Y)
+        ( map-isometry-cauchy-approximation-Pseudometric-Space X Y f)
+    pr1
+      ( is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+        d (x , is-approx-x) (y , is-approx-y))
+      Ndxy δ ε =
+      preserves-neighborhoods-map-isometry-Pseudometric-Space
+        ( X)
+        ( Y)
+        ( f)
+        ( δ +ℚ⁺ ε +ℚ⁺ d)
+        ( x δ)
+        ( y ε)
+        ( Ndxy δ ε)
+    pr2
+      ( is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+        d (x , is-approx-x) (y , is-approx-y))
+      Ndfxfy δ ε =
+      reflects-neighborhoods-map-isometry-Pseudometric-Space
+        ( X)
+        ( Y)
+        ( f)
+        ( δ +ℚ⁺ ε +ℚ⁺ d)
+        ( x δ)
+        ( y ε)
+        ( Ndfxfy δ ε)
+
+  isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space X)
+      ( cauchy-pseudocompletion-Pseudometric-Space Y)
+  isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space =
+    ( map-isometry-cauchy-approximation-Pseudometric-Space X Y f ,
+      is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space)
+```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
new file mode 100644
index 00000000000..281e54b3bd9
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
@@ -0,0 +1,92 @@
+# The action on Cauchy approximations of short maps in pseudometric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-approximations-pseudometric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
+open import metric-spaces.pseudometric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The action of [short maps](metric-spaces.short-maps-pseudometric-spaces.md) on
+[pseudometric spaces](metric-spaces.pseudometric-spaces.md) preserves
+[Cauchy approximations](metric-spaces.cauchy-approximations-pseudometric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
+  (x@(map-x , is-approx-x) : cauchy-approximation-Pseudometric-Space X)
+  where
+
+  map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space :
+    ℚ⁺ → type-Pseudometric-Space Y
+  map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space
+    ε =
+    map-f (map-x ε)
+
+  abstract
+    is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space :
+      is-cauchy-approximation-Pseudometric-Space
+        ( Y)
+        ( map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space)
+    is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space
+      δ ε =
+      is-short-f (δ +ℚ⁺ ε) (map-x δ) (map-x ε) (is-approx-x δ ε)
+
+  map-short-map-cauchy-approximation-Pseudometric-Space :
+    cauchy-approximation-Pseudometric-Space Y
+  map-short-map-cauchy-approximation-Pseudometric-Space =
+    ( map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space ,
+      is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space)
+```
+
+## Properties
+
+### The induced action of a short map on Cauchy approximations is short in the Cauchy pseudocompletion of a pseudometric space
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Pseudometric-Space l1 l2)
+  (Y : Pseudometric-Space l3 l4)
+  (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
+  where
+
+  abstract
+    is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space :
+      is-short-map-Pseudometric-Space
+        ( cauchy-pseudocompletion-Pseudometric-Space X)
+        ( cauchy-pseudocompletion-Pseudometric-Space Y)
+        ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f)
+    is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space
+      d (x , _) (y , _) Ndxy δ ε =
+      is-short-f (δ +ℚ⁺ ε +ℚ⁺ d) (x δ) (y ε) (Ndxy δ ε)
+
+  short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space :
+    short-map-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space X)
+      ( cauchy-pseudocompletion-Pseudometric-Space Y)
+  short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space =
+    ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f ,
+      is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space)
+```
diff --git a/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md b/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md
index b434d07f104..edb2a44f4f2 100644
--- a/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-pseudocompletions-of-metric-spaces.lagda.md
@@ -13,6 +13,7 @@ open import elementary-number-theory.positive-rational-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
+open import foundation.equivalence-relations
 open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
@@ -253,3 +254,57 @@ module _
     isometry-lim-cauchy-approximation-cauchy-pseudocompletion-Pseudometric-Space
       ( pseudometric-Metric-Space M)
 ```
+
+### The similarity relation in a Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  where
+
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space :
+    equivalence-relation l2 (cauchy-approximation-Metric-Space M)
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Space =
+    equivalence-relation-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space M)
+
+  sim-prop-cauchy-pseudocompletion-Metric-Space :
+    Relation-Prop l2 (cauchy-approximation-Metric-Space M)
+  sim-prop-cauchy-pseudocompletion-Metric-Space =
+    sim-prop-Pseudometric-Space (cauchy-pseudocompletion-Metric-Space M)
+
+  sim-cauchy-pseudocompletion-Metric-Space :
+    Relation l2 (cauchy-approximation-Metric-Space M)
+  sim-cauchy-pseudocompletion-Metric-Space =
+    type-Relation-Prop sim-prop-cauchy-pseudocompletion-Metric-Space
+```
+
+### If two constant Cauchy approximations are similar, they have the same constant
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  where abstract
+
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Space :
+    (x y : type-Metric-Space M) →
+    sim-cauchy-pseudocompletion-Metric-Space M
+      ( const-cauchy-approximation-Metric-Space M x)
+      ( const-cauchy-approximation-Metric-Space M y) →
+    x ＝ y
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Space
+    x y cx~cy =
+    eq-sim-Metric-Space
+      ( M)
+      ( x)
+      ( y)
+      ( λ ε →
+        let
+          (ε12 , ε3 , ε12+ε3=ε) = split-ℚ⁺ ε
+          (ε1 , ε2 , ε1+ε2=ε12) = split-ℚ⁺ ε12
+        in
+          tr
+            ( λ d → neighborhood-Metric-Space M d x y)
+            ( ap-add-ℚ⁺ ε1+ε2=ε12 refl ∙ ε12+ε3=ε)
+            ( cx~cy ε3 ε1 ε2))
+```
diff --git a/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md b/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
index 7b9c060eed8..205625ab962 100644
--- a/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
@@ -19,6 +19,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.dependent-products-propositions
+open import foundation.equivalence-relations
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
@@ -279,6 +280,30 @@ module _
       is-saturated-neighborhood-cauchy-pseudocompletion-Pseudometric-Space M)
 ```
 
+### The similarity equivalence relation in the Cauchy pseudocompletion of a pseudometric space
+
+```agda
+module _
+  {l1 l2 : Level} (M : Pseudometric-Space l1 l2)
+  where
+
+  equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space :
+    equivalence-relation l2 (cauchy-approximation-Pseudometric-Space M)
+  equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space =
+    equivalence-relation-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Pseudometric-Space M)
+
+  sim-prop-cauchy-pseudocompletion-Pseudometric-Space :
+    Relation-Prop l2 (cauchy-approximation-Pseudometric-Space M)
+  sim-prop-cauchy-pseudocompletion-Pseudometric-Space =
+    sim-prop-Pseudometric-Space (cauchy-pseudocompletion-Pseudometric-Space M)
+
+  sim-cauchy-pseudocompletion-Pseudometric-Space :
+    Relation l2 (cauchy-approximation-Pseudometric-Space M)
+  sim-cauchy-pseudocompletion-Pseudometric-Space =
+    sim-Pseudometric-Space (cauchy-pseudocompletion-Pseudometric-Space M)
+```
+
 ### The isometry from a pseudometric space to its Cauchy pseudocompletion
 
 ```agda

From a3b8c80684efc1f6aa855a723be5f904a6ffbac7 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:17:55 -0700
Subject: [PATCH 04/26] Progress

---
 ...roximations-metric-abelian-groups.lagda.md | 456 ++++++++++++++++++
 ...roximations-metric-abelian-groups.lagda.md |  70 +++
 ...completions-metric-abelian-groups.lagda.md |  94 ++++
 src/analysis/metric-abelian-groups.lagda.md   | 157 +++++-
 ...quality-positive-rational-numbers.lagda.md |  13 +
 5 files changed, 773 insertions(+), 17 deletions(-)
 create mode 100644 src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
 create mode 100644 src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md
 create mode 100644 src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md

diff --git a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..15b5c86afe3
--- /dev/null
+++ b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -0,0 +1,456 @@
+# Addition of Cauchy approximations in metric abelian groups
+
+```agda
+module analysis.addition-cauchy-approximations-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.strict-inequality-positive-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.binary-functoriality-set-quotients
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.metric-quotients-of-pseudometric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+[Cauchy approximations](analysis.cauchy-approximations-metric-abelian-groups.md)
+in [metric abelian groups](analysis.metric-abelian-groups.md) admit an addition
+operation whose properties resemble an
+[abelian group](group-theory.abelian-groups.md) with respect to the
+[similarity relationship](metric-spaces.similarity-of-elements-pseudometric-spaces.md)
+of the
+[Cauchy pseudocompletion of the metric abelian group](analysis.cauchy-pseudocompletions-metric-abelian-groups.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  ((x , is-approx-x) (y , is-approx-y) :
+    cauchy-approximation-Metric-Ab G)
+  where
+
+  opaque
+    map-add-cauchy-approximation-Metric-Ab :
+      ℚ⁺ → type-Metric-Ab G
+    map-add-cauchy-approximation-Metric-Ab ε =
+      let (δ , _) = bound-double-le-ℚ⁺ ε in add-Metric-Ab G (x δ) (y δ)
+
+  abstract opaque
+    unfolding map-add-cauchy-approximation-Metric-Ab
+
+    is-cauchy-approximation-map-add-cauchy-approximation-Metric-Ab :
+      is-cauchy-approximation-Metric-Ab G map-add-cauchy-approximation-Metric-Ab
+    is-cauchy-approximation-map-add-cauchy-approximation-Metric-Ab δ ε =
+      let
+        (δ' , 2δ'<δ) = bound-double-le-ℚ⁺ δ
+        (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      in
+        monotonic-neighborhood-Metric-Ab G
+          ( add-Metric-Ab G (x δ') (y δ'))
+          ( add-Metric-Ab G (x ε') (y ε'))
+          ( (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε'))
+          ( δ +ℚ⁺ ε)
+          ( concat-eq-le-ℚ⁺
+            { z = δ +ℚ⁺ ε}
+            ( interchange-law-add-add-ℚ⁺ δ' ε' δ' ε')
+            ( preserves-le-add-ℚ 2δ'<δ 2ε'<ε))
+          ( neighborhood-add-Metric-Ab
+            ( G)
+            ( δ' +ℚ⁺ ε')
+            ( δ' +ℚ⁺ ε')
+            ( x δ')
+            ( x ε')
+            ( y δ')
+            ( y ε')
+            ( is-approx-x δ' ε')
+            ( is-approx-y δ' ε'))
+
+  add-cauchy-approximation-Metric-Ab : cauchy-approximation-Metric-Ab G
+  add-cauchy-approximation-Metric-Ab =
+    ( map-add-cauchy-approximation-Metric-Ab ,
+      is-cauchy-approximation-map-add-cauchy-approximation-Metric-Ab)
+```
+
+## Properties
+
+### Addition of Cauchy approximations is a similarity-preserving binary map
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  preserves-sim-add-cauchy-approximation-Metric-Ab :
+    preserves-sim-binary-map-equivalence-relation
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
+      ( add-cauchy-approximation-Metric-Ab G)
+  preserves-sim-add-cauchy-approximation-Metric-Ab
+    {x , is-approx-x} {x' , is-approx-x'} {y , is-approx-y} {y' , is-approx-y'}
+    x~x' y~y' δ ε θ =
+    let
+      (δ' , 2δ'<δ) = bound-double-le-ℚ⁺ δ
+      (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      (θ' , 2θ'<θ) = bound-double-le-ℚ⁺ θ
+    in
+      monotonic-neighborhood-Metric-Ab G
+        ( add-Metric-Ab G (x ε') (y ε'))
+        ( add-Metric-Ab G (x' θ') (y' θ'))
+        ( (ε' +ℚ⁺ θ' +ℚ⁺ δ') +ℚ⁺ (ε' +ℚ⁺ θ' +ℚ⁺ δ'))
+        ( ε +ℚ⁺ θ +ℚ⁺ δ)
+        ( concat-eq-le-ℚ⁺
+          { z = ε +ℚ⁺ θ +ℚ⁺ δ}
+          ( equational-reasoning
+            (ε' +ℚ⁺ θ' +ℚ⁺ δ') +ℚ⁺ (ε' +ℚ⁺ θ' +ℚ⁺ δ')
+            ＝ ((ε' +ℚ⁺ θ') +ℚ⁺ (ε' +ℚ⁺ θ')) +ℚ⁺ (δ' +ℚ⁺ δ')
+              by interchange-law-add-add-ℚ⁺ _ _ _ _
+            ＝ (ε' +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ θ') +ℚ⁺ (δ' +ℚ⁺ δ')
+              by ap-add-ℚ⁺ (interchange-law-add-add-ℚ⁺ ε' θ' ε' θ') refl)
+          ( preserves-le-add-ℚ
+            ( preserves-le-add-ℚ 2ε'<ε 2θ'<θ)
+            ( 2δ'<δ)))
+        ( neighborhood-add-Metric-Ab
+          ( G)
+          ( ε' +ℚ⁺ θ' +ℚ⁺ δ')
+          ( ε' +ℚ⁺ θ' +ℚ⁺ δ')
+          ( x ε')
+          ( x' θ')
+          ( y ε')
+          ( y' θ')
+          ( x~x' δ' ε' θ')
+          ( y~y' δ' ε' θ'))
+```
+
+### The addition of two constant Cauchy approximations for `x` and `y` is similar to the constant approximation for `x + y`
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  sim-add-const-cauchy-approximation-Metric-Ab :
+    (x y : type-Metric-Ab G) →
+    sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space
+        ( metric-space-Metric-Ab G))
+      ( add-cauchy-approximation-Metric-Ab G
+        ( const-cauchy-approximation-Metric-Ab G x)
+        ( const-cauchy-approximation-Metric-Ab G y))
+      ( const-cauchy-approximation-Metric-Ab G (add-Metric-Ab G x y))
+  sim-add-const-cauchy-approximation-Metric-Ab x y δ ε θ =
+    refl-neighborhood-Metric-Ab G (ε +ℚ⁺ θ +ℚ⁺ δ) (add-Metric-Ab G x y)
+```
+
+### Addition is associative relative to the similarity relation
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) ay@(y , is-approx-y) az@(z , is-approx-z) :
+    cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  sim-associative-add-cauchy-approximation-Metric-Ab :
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( add-cauchy-approximation-Metric-Ab G
+        ( add-cauchy-approximation-Metric-Ab G ax ay)
+        ( az))
+      ( add-cauchy-approximation-Metric-Ab G
+        ( ax)
+        ( add-cauchy-approximation-Metric-Ab G ay az))
+  sim-associative-add-cauchy-approximation-Metric-Ab δ ε θ =
+    let
+      (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      (ε'' , 2ε''<ε') = bound-double-le-ℚ⁺ ε'
+      (θ' , 2θ'<θ) = bound-double-le-ℚ⁺ θ
+      (θ'' , 2θ''<θ') = bound-double-le-ℚ⁺ θ'
+      xyz1 = add-Metric-Ab G (add-Metric-Ab G (x ε'') (y ε'')) (z ε')
+      xyz2 = add-Metric-Ab G (add-Metric-Ab G (x θ') (y θ'')) (z θ'')
+    in
+      tr
+        ( neighborhood-Metric-Ab G (ε +ℚ⁺ θ +ℚ⁺ δ) xyz1)
+        ( associative-add-Metric-Ab G _ _ _)
+        ( monotonic-neighborhood-Metric-Ab G
+          ( xyz1)
+          ( xyz2)
+          ( (ε'' +ℚ⁺ θ') +ℚ⁺ (ε'' +ℚ⁺ θ'') +ℚ⁺ (ε' +ℚ⁺ θ''))
+          ( ε +ℚ⁺ θ +ℚ⁺ δ)
+          ( concat-eq-le-ℚ⁺
+            { z = ε +ℚ⁺ θ +ℚ⁺ δ}
+            ( equational-reasoning
+              (ε'' +ℚ⁺ θ') +ℚ⁺ (ε'' +ℚ⁺ θ'') +ℚ⁺ (ε' +ℚ⁺ θ'')
+              ＝ ((ε'' +ℚ⁺ ε'') +ℚ⁺ (θ' +ℚ⁺ θ'')) +ℚ⁺ (ε' +ℚ⁺ θ'')
+                by ap-add-ℚ⁺ (interchange-law-add-add-ℚ⁺ _ _ _ _) refl
+              ＝ ((ε'' +ℚ⁺ ε'') +ℚ⁺ ε') +ℚ⁺ ((θ' +ℚ⁺ θ'') +ℚ⁺ θ'')
+                by interchange-law-add-add-ℚ⁺ _ _ _ _
+              ＝ ((ε'' +ℚ⁺ ε'') +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ (θ'' +ℚ⁺ θ''))
+                by ap-add-ℚ⁺ refl (associative-add-ℚ⁺ _ _ _))
+            ( transitive-le-ℚ⁺
+              ( ((ε'' +ℚ⁺ ε'') +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ (θ'' +ℚ⁺ θ'')))
+              ( (ε' +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ θ'))
+              ( ε +ℚ⁺ θ +ℚ⁺ δ)
+              ( transitive-le-ℚ⁺
+                ( (ε' +ℚ⁺ ε') +ℚ⁺ (θ' +ℚ⁺ θ'))
+                ( ε +ℚ⁺ θ)
+                ( ε +ℚ⁺ θ +ℚ⁺ δ)
+                ( le-left-add-ℚ⁺ (ε +ℚ⁺ θ) δ)
+                ( preserves-le-add-ℚ 2ε'<ε 2θ'<θ))
+              ( preserves-le-add-ℚ
+                ( preserves-le-left-add-ℚ _ _ _ 2ε''<ε')
+                ( preserves-le-right-add-ℚ _ _ _ 2θ''<θ'))))
+          ( neighborhood-add-Metric-Ab G
+            ( (ε'' +ℚ⁺ θ') +ℚ⁺ (ε'' +ℚ⁺ θ''))
+            ( ε' +ℚ⁺ θ'')
+            ( add-Metric-Ab G (x ε'') (y ε''))
+            ( add-Metric-Ab G (x θ') (y θ''))
+            ( z ε')
+            ( z θ'')
+            ( neighborhood-add-Metric-Ab G
+              ( ε'' +ℚ⁺ θ')
+              ( ε'' +ℚ⁺ θ'')
+              ( x ε'')
+              ( x θ')
+              ( y ε'')
+              ( y θ'')
+              ( is-approx-x ε'' θ')
+              ( is-approx-y ε'' θ''))
+            ( is-approx-z ε' θ'')))
+```
+
+### Commutativity of addition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (x y : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  commutative-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G x y ＝
+    add-cauchy-approximation-Metric-Ab G y x
+  commutative-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → commutative-add-Metric-Ab G _ _))
+```
+
+### Unit laws
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  sim-left-unit-law-add-cauchy-approximation-Metric-Ab :
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( add-cauchy-approximation-Metric-Ab G
+        ( zero-cauchy-approximation-Metric-Ab G)
+        ( ax))
+      ( ax)
+  sim-left-unit-law-add-cauchy-approximation-Metric-Ab δ ε θ =
+    let (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε in
+    monotonic-neighborhood-Metric-Ab G
+      ( add-Metric-Ab G (zero-Metric-Ab G) (x ε'))
+      ( x θ)
+      ( ε' +ℚ⁺ θ)
+      ( ε +ℚ⁺ θ +ℚ⁺ δ)
+      ( transitive-le-ℚ⁺
+        ( ε' +ℚ⁺ θ)
+        ( ε +ℚ⁺ θ)
+        ( ε +ℚ⁺ θ +ℚ⁺ δ)
+        ( le-left-add-ℚ⁺ (ε +ℚ⁺ θ) δ)
+        ( preserves-le-left-add-ℚ _ _ _ (le-modulus-le-double-le-ℚ⁺ ε)))
+      ( inv-tr
+        ( λ y → neighborhood-Metric-Ab G (ε' +ℚ⁺ θ) y (x θ))
+        ( left-unit-law-add-Metric-Ab G (x ε'))
+        ( is-approx-x ε' θ))
+
+  sim-right-unit-law-add-cauchy-approximation-Metric-Ab :
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( add-cauchy-approximation-Metric-Ab G
+        ( ax)
+        ( zero-cauchy-approximation-Metric-Ab G))
+      ( ax)
+  sim-right-unit-law-add-cauchy-approximation-Metric-Ab =
+    tr
+      ( λ ay → sim-cauchy-pseudocompletion-Metric-Ab G ay ax)
+      ( commutative-add-cauchy-approximation-Metric-Ab G
+        ( zero-cauchy-approximation-Metric-Ab G)
+        ( ax))
+      ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab)
+```
+
+### Negations of Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  neg-cauchy-approximation-Metric-Ab :
+    cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
+  neg-cauchy-approximation-Metric-Ab =
+    map-isometry-cauchy-approximation-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-space-Metric-Ab G)
+      ( isometry-neg-Metric-Ab G)
+```
+
+### Inverse laws of Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  left-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( neg-cauchy-approximation-Metric-Ab G ax)
+      ( ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  left-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → left-inverse-law-add-Metric-Ab G _))
+
+  right-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( ax)
+      ( neg-cauchy-approximation-Metric-Ab G ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  right-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → right-inverse-law-add-Metric-Ab G _))
+```
+
+### Negations of Cauchy approximations preserve similarity
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  preserves-sim-neg-cauchy-approximation-Metric-Ab :
+    (x y : cauchy-approximation-Metric-Ab G) →
+    sim-cauchy-pseudocompletion-Metric-Ab G x y →
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( neg-cauchy-approximation-Metric-Ab G x)
+      ( neg-cauchy-approximation-Metric-Ab G y)
+  preserves-sim-neg-cauchy-approximation-Metric-Ab =
+    preserves-sim-map-isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( isometry-cauchy-pseudocompletion-isometry-Metric-Space
+        ( metric-space-Metric-Ab G)
+        ( metric-space-Metric-Ab G)
+        ( isometry-neg-Metric-Ab G))
+```
+
+### Left addition preserves neighborhoods
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  abstract opaque
+    unfolding map-add-cauchy-approximation-Metric-Ab
+
+    preserves-neighborhood-left-add-cauchy-approximation-Metric-Ab :
+      (x : cauchy-approximation-Metric-Ab G) →
+      is-short-map-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( add-cauchy-approximation-Metric-Ab G x)
+    preserves-neighborhood-left-add-cauchy-approximation-Metric-Ab
+      (x , is-approx-x) d (y , is-approx-y) (z , is-approx-z) Ndyz δ ε =
+      let
+        (δ' , 2δ'<δ) = bound-double-le-ℚ⁺ δ
+        (ε' , 2ε'<ε) = bound-double-le-ℚ⁺ ε
+      in
+        monotonic-neighborhood-Metric-Ab G
+          ( add-Metric-Ab G (x δ') (y δ'))
+          ( add-Metric-Ab G (x ε') (z ε'))
+          ( (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε' +ℚ⁺ d))
+          ( δ +ℚ⁺ ε +ℚ⁺ d)
+          ( concat-eq-le-ℚ⁺
+            { z = δ +ℚ⁺ ε +ℚ⁺ d}
+            ( equational-reasoning
+              (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε' +ℚ⁺ d)
+              ＝ (δ' +ℚ⁺ ε') +ℚ⁺ (δ' +ℚ⁺ ε') +ℚ⁺ d
+                by inv (associative-add-ℚ⁺ _ _ _)
+              ＝ (δ' +ℚ⁺ δ') +ℚ⁺ (ε' +ℚ⁺ ε') +ℚ⁺ d
+                by ap-add-ℚ⁺ (interchange-law-add-add-ℚ⁺ _ _ _ _) refl)
+            ( preserves-le-left-add-ℚ _ _ _ (preserves-le-add-ℚ 2δ'<δ 2ε'<ε)))
+          ( neighborhood-add-Metric-Ab G
+            ( δ' +ℚ⁺ ε')
+            ( δ' +ℚ⁺ ε' +ℚ⁺ d)
+            ( x δ')
+            ( x ε')
+            ( y δ')
+            ( z ε')
+            ( is-approx-x δ' ε')
+            ( Ndyz δ' ε'))
+
+  short-map-left-add-cauchy-pseudocompletion-Metric-Ab :
+    (x : cauchy-approximation-Metric-Ab G) →
+    short-map-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+  short-map-left-add-cauchy-pseudocompletion-Metric-Ab x =
+    ( add-cauchy-approximation-Metric-Ab G x ,
+      preserves-neighborhood-left-add-cauchy-approximation-Metric-Ab x)
+```
diff --git a/src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..f93cdc3d811
--- /dev/null
+++ b/src/analysis/cauchy-approximations-metric-abelian-groups.lagda.md
@@ -0,0 +1,70 @@
+# Cauchy approximations in metric abelian groups
+
+```agda
+module analysis.cauchy-approximations-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-approximations-metric-spaces
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "Cauchy approximation" Disambiguation="in a metric abelian group" Agda=cauchy-approximation-Metric-Ab}}
+in a [metric abelian group](analysis.metric-abelian-groups.md) is a
+[Cauchy approximation](metric-spaces.cauchy-approximations-metric-spaces.md) in
+the underlying [metric space](metric-spaces.metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  is-cauchy-approximation-prop-Metric-Ab : subtype l2 (ℚ⁺ → type-Metric-Ab G)
+  is-cauchy-approximation-prop-Metric-Ab =
+    is-cauchy-approximation-prop-Metric-Space (metric-space-Metric-Ab G)
+
+  is-cauchy-approximation-Metric-Ab : (ℚ⁺ → type-Metric-Ab G) → UU l2
+  is-cauchy-approximation-Metric-Ab =
+    is-in-subtype is-cauchy-approximation-prop-Metric-Ab
+
+  cauchy-approximation-Metric-Ab : UU (l1 ⊔ l2)
+  cauchy-approximation-Metric-Ab =
+    type-subtype is-cauchy-approximation-prop-Metric-Ab
+```
+
+## Properties
+
+### Constant maps in metric abelian groups are Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  const-cauchy-approximation-Metric-Ab :
+    type-Metric-Ab G → cauchy-approximation-Metric-Ab G
+  const-cauchy-approximation-Metric-Ab =
+    const-cauchy-approximation-Metric-Space (metric-space-Metric-Ab G)
+
+  zero-cauchy-approximation-Metric-Ab :
+    cauchy-approximation-Metric-Ab G
+  zero-cauchy-approximation-Metric-Ab =
+    const-cauchy-approximation-Metric-Ab (zero-Metric-Ab G)
+```
diff --git a/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..63b1685f2e2
--- /dev/null
+++ b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -0,0 +1,94 @@
+# Cauchy pseudocompletions of metric abelian groups
+
+```agda
+module analysis.cauchy-pseudocompletions-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.binary-relations
+open import foundation.equivalence-relations
+open import foundation.identity-types
+open import foundation.universe-levels
+
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.pseudometric-spaces
+open import metric-spaces.rational-neighborhood-relations
+open import metric-spaces.similarity-of-elements-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "Cauchy pseudocompletion" Disambiguation="of a metric abelian group" Agda=cauchy-pseudocompletion-Metric-Ab}}
+of a [metric abelian group](group-theory.metric-abelian-groups.md) is the
+[Cauchy pseudocompletion](metric-spaces.cauchy-pseudocompletions-of-metric-spaces.md)
+of the underlying [metric space](metric-spaces.metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  cauchy-pseudocompletion-Metric-Ab : Pseudometric-Space (l1 ⊔ l2) l2
+  cauchy-pseudocompletion-Metric-Ab =
+    cauchy-pseudocompletion-Metric-Space (metric-space-Metric-Ab G)
+
+  neighborhood-prop-cauchy-pseudocompletion-Metric-Ab :
+    Rational-Neighborhood-Relation l2 (cauchy-approximation-Metric-Ab G)
+  neighborhood-prop-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-prop-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+
+  neighborhood-cauchy-pseudocompletion-Metric-Ab :
+    ℚ⁺ → Relation l2 (cauchy-approximation-Metric-Ab G)
+  neighborhood-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab :
+    equivalence-relation l2 (cauchy-approximation-Metric-Ab G)
+  equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab =
+    equivalence-relation-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab)
+
+  sim-prop-cauchy-pseudocompletion-Metric-Ab :
+    Relation-Prop l2 (cauchy-approximation-Metric-Ab G)
+  sim-prop-cauchy-pseudocompletion-Metric-Ab =
+    sim-prop-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+
+  sim-cauchy-pseudocompletion-Metric-Ab :
+    Relation l2 (cauchy-approximation-Metric-Ab G)
+  sim-cauchy-pseudocompletion-Metric-Ab =
+    sim-Pseudometric-Space cauchy-pseudocompletion-Metric-Ab
+```
+
+## Properties
+
+### If two constant Cauchy approximations are similar, they have the same constant
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where abstract
+
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab :
+    (x y : type-Metric-Ab G) →
+    sim-cauchy-pseudocompletion-Metric-Ab G
+      ( const-cauchy-approximation-Metric-Ab G x)
+      ( const-cauchy-approximation-Metric-Ab G y) →
+    x ＝ y
+  eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab =
+    eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Space
+      ( metric-space-Metric-Ab G)
+```
diff --git a/src/analysis/metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups.lagda.md
index bf9a4eaf0c3..57eaaa6771c 100644
--- a/src/analysis/metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups.lagda.md
@@ -7,16 +7,19 @@ module analysis.metric-abelian-groups where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.binary-relations
+open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.dependent-products-propositions
 open import foundation.function-extensionality
 open import foundation.identity-types
+open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -26,11 +29,15 @@ open import group-theory.abelian-groups
 open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.extensionality-pseudometric-spaces
 open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-maps-metric-spaces
+open import metric-spaces.monotonic-rational-neighborhood-relations
 open import metric-spaces.pseudometric-spaces
 open import metric-spaces.rational-neighborhood-relations
+open import metric-spaces.reflexive-rational-neighborhood-relations
+open import metric-spaces.short-maps-metric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
+open import metric-spaces.triangular-rational-neighborhood-relations
 open import metric-spaces.uniformly-continuous-maps-metric-spaces
 ```
 
@@ -41,8 +48,9 @@ open import metric-spaces.uniformly-continuous-maps-metric-spaces
 A {{#concept "metric abelian group" Agda=Metric-Ab}} is an
 [abelian group](group-theory.abelian-groups.md) endowed with the structure of a
 [metric space](metric-spaces.metric-spaces.md) such that the addition operation
-and negation operation are
-[isometries](metric-spaces.isometries-metric-spaces.md).
+and negation operation are [short](metric-spaces.short-maps-metric-spaces.md)
+(which, together with the group operations, implies they are
+[isometries](metric-spaces.isometries-metric-spaces.md)).
 
 ## Definition
 
@@ -55,10 +63,10 @@ is-metric-ab-prop-Ab-Pseudometric-Structure G M =
     MS = (type-Ab G , M)
   in
     is-extensional-prop-Pseudometric-Space MS ∧
-    is-isometry-prop-Pseudometric-Space MS MS (neg-Ab G) ∧
+    is-short-map-prop-Pseudometric-Space MS MS (neg-Ab G) ∧
     Π-Prop
       ( type-Ab G)
-      ( λ x → is-isometry-prop-Pseudometric-Space MS MS (add-Ab G x))
+      ( λ x → is-short-map-prop-Pseudometric-Space MS MS (add-Ab G x))
 
 is-metric-ab-Ab-Pseudometric-Structure :
   {l1 l2 : Level} (G : Ab l1) (M : Pseudometric-Structure l2 (type-Ab G)) →
@@ -89,32 +97,52 @@ module _
 ```agda
 module _
   {l1 l2 : Level} (MG : Metric-Ab l1 l2)
+  (let ab-MG = ab-Metric-Ab MG)
   where
 
   zero-Metric-Ab : type-Metric-Ab MG
-  zero-Metric-Ab = zero-Ab (ab-Metric-Ab MG)
+  zero-Metric-Ab = zero-Ab ab-MG
 
   add-Metric-Ab : type-Metric-Ab MG → type-Metric-Ab MG → type-Metric-Ab MG
-  add-Metric-Ab = add-Ab (ab-Metric-Ab MG)
+  add-Metric-Ab = add-Ab ab-MG
 
   add-Metric-Ab' : type-Metric-Ab MG → type-Metric-Ab MG → type-Metric-Ab MG
-  add-Metric-Ab' = add-Ab' (ab-Metric-Ab MG)
+  add-Metric-Ab' = add-Ab' ab-MG
 
   ap-add-Metric-Ab :
     {x x' y y' : type-Metric-Ab MG} → x ＝ x' → y ＝ y' →
     add-Metric-Ab x y ＝ add-Metric-Ab x' y'
-  ap-add-Metric-Ab = ap-add-Ab (ab-Metric-Ab MG)
+  ap-add-Metric-Ab = ap-add-Ab ab-MG
 
   neg-Metric-Ab : type-Metric-Ab MG → type-Metric-Ab MG
-  neg-Metric-Ab = neg-Ab (ab-Metric-Ab MG)
+  neg-Metric-Ab = neg-Ab ab-MG
 
   abstract
+    left-unit-law-add-Metric-Ab :
+      (x : type-Metric-Ab MG) → add-Metric-Ab zero-Metric-Ab x ＝ x
+    left-unit-law-add-Metric-Ab = left-unit-law-add-Ab ab-MG
+
+    associative-add-Metric-Ab :
+      (x y z : type-Metric-Ab MG) →
+      add-Metric-Ab (add-Metric-Ab x y) z ＝ add-Metric-Ab x (add-Metric-Ab y z)
+    associative-add-Metric-Ab = associative-add-Ab ab-MG
+
+    left-inverse-law-add-Metric-Ab :
+      (x : type-Metric-Ab MG) →
+      add-Metric-Ab (neg-Metric-Ab x) x ＝ zero-Metric-Ab
+    left-inverse-law-add-Metric-Ab = left-inverse-law-add-Ab ab-MG
+
+    right-inverse-law-add-Metric-Ab :
+      (x : type-Metric-Ab MG) →
+      add-Metric-Ab x (neg-Metric-Ab x) ＝ zero-Metric-Ab
+    right-inverse-law-add-Metric-Ab = right-inverse-law-add-Ab ab-MG
+
     neg-zero-Metric-Ab : neg-Metric-Ab zero-Metric-Ab ＝ zero-Metric-Ab
-    neg-zero-Metric-Ab = neg-zero-Ab (ab-Metric-Ab MG)
+    neg-zero-Metric-Ab = neg-zero-Ab ab-MG
 
     neg-neg-Metric-Ab :
       (x : type-Metric-Ab MG) → neg-Metric-Ab (neg-Metric-Ab x) ＝ x
-    neg-neg-Metric-Ab = neg-neg-Ab (ab-Metric-Ab MG)
+    neg-neg-Metric-Ab = neg-neg-Ab ab-MG
 
   diff-Metric-Ab : type-Metric-Ab MG → type-Metric-Ab MG → type-Metric-Ab MG
   diff-Metric-Ab x y = add-Metric-Ab x (neg-Metric-Ab y)
@@ -122,16 +150,16 @@ module _
   ap-diff-Metric-Ab :
     {x x' y y' : type-Metric-Ab MG} → x ＝ x' → y ＝ y' →
     diff-Metric-Ab x y ＝ diff-Metric-Ab x' y'
-  ap-diff-Metric-Ab = ap-right-subtraction-Ab (ab-Metric-Ab MG)
+  ap-diff-Metric-Ab = ap-right-subtraction-Ab ab-MG
 
   commutative-add-Metric-Ab :
     (x y : type-Metric-Ab MG) → add-Metric-Ab x y ＝ add-Metric-Ab y x
-  commutative-add-Metric-Ab = commutative-add-Ab (ab-Metric-Ab MG)
+  commutative-add-Metric-Ab = commutative-add-Ab ab-MG
 
   is-identity-right-conjugation-Metric-Ab :
     (x y : type-Metric-Ab MG) → add-Metric-Ab x (diff-Metric-Ab y x) ＝ y
   is-identity-right-conjugation-Metric-Ab =
-    is-identity-right-conjugation-Ab (ab-Metric-Ab MG)
+    is-identity-right-conjugation-Ab ab-MG
 ```
 
 ### Metric properties of metric abelian groups
@@ -162,13 +190,55 @@ module _
   neighborhood-Metric-Ab : ℚ⁺ → Relation l2 (type-Metric-Ab MG)
   neighborhood-Metric-Ab = neighborhood-Metric-Space metric-space-Metric-Ab
 
+  refl-neighborhood-Metric-Ab :
+    is-reflexive-Rational-Neighborhood-Relation neighborhood-prop-Metric-Ab
+  refl-neighborhood-Metric-Ab =
+    refl-neighborhood-Metric-Space metric-space-Metric-Ab
+
+  monotonic-neighborhood-Metric-Ab :
+    is-monotonic-Rational-Neighborhood-Relation neighborhood-prop-Metric-Ab
+  monotonic-neighborhood-Metric-Ab =
+    monotonic-neighborhood-Metric-Space metric-space-Metric-Ab
+
+  triangular-neighborhood-Metric-Ab :
+    is-triangular-Rational-Neighborhood-Relation neighborhood-prop-Metric-Ab
+  triangular-neighborhood-Metric-Ab =
+    triangular-neighborhood-Metric-Space metric-space-Metric-Ab
+
+  is-short-map-add-Metric-Ab :
+    (x : type-Metric-Ab MG) →
+    is-short-map-Metric-Space
+      ( metric-space-Metric-Ab)
+      ( metric-space-Metric-Ab)
+      ( add-Metric-Ab MG x)
+  is-short-map-add-Metric-Ab = pr2 (pr2 (pr2 (pr2 MG)))
+
+  abstract
+    reflects-neighborhoods-left-add-Metric-Ab :
+      (x : type-Metric-Ab MG)
+      (d : ℚ⁺)
+      (y z : type-Metric-Ab MG) →
+      neighborhood-Metric-Ab
+        ( d)
+        ( add-Metric-Ab MG x y)
+        ( add-Metric-Ab MG x z) →
+      neighborhood-Metric-Ab d y z
+    reflects-neighborhoods-left-add-Metric-Ab x d y z Nd⟨x+y⟩⟨x+z⟩ =
+      binary-tr
+        ( neighborhood-Metric-Ab d)
+        ( is-retraction-left-subtraction-Ab (ab-Metric-Ab MG) x y)
+        ( is-retraction-left-subtraction-Ab (ab-Metric-Ab MG) x z)
+        ( is-short-map-add-Metric-Ab (neg-Metric-Ab MG x) d _ _ Nd⟨x+y⟩⟨x+z⟩)
+
   is-isometry-add-Metric-Ab :
     (x : type-Metric-Ab MG) →
     is-isometry-Metric-Space
       ( metric-space-Metric-Ab)
       ( metric-space-Metric-Ab)
       ( add-Metric-Ab MG x)
-  is-isometry-add-Metric-Ab = pr2 (pr2 (pr2 (pr2 MG)))
+  is-isometry-add-Metric-Ab x d y z =
+    ( is-short-map-add-Metric-Ab x d y z ,
+      reflects-neighborhoods-left-add-Metric-Ab x d y z)
 
   isometry-add-Metric-Ab :
     (x : type-Metric-Ab MG) →
@@ -200,12 +270,33 @@ module _
   isometry-add-Metric-Ab' x =
     ( add-Metric-Ab' MG x , is-isometry-add-Metric-Ab' x)
 
+  is-short-map-neg-Metric-Ab :
+    is-short-map-Metric-Space
+      ( metric-space-Metric-Ab)
+      ( metric-space-Metric-Ab)
+      ( neg-Metric-Ab MG)
+  is-short-map-neg-Metric-Ab = pr1 (pr2 (pr2 (pr2 MG)))
+
+  abstract
+    reflects-neighborhoods-neg-Metric-Ab :
+      (d : ℚ⁺) (x y : type-Metric-Ab MG) →
+      neighborhood-Metric-Ab d (neg-Metric-Ab MG x) (neg-Metric-Ab MG y) →
+      neighborhood-Metric-Ab d x y
+    reflects-neighborhoods-neg-Metric-Ab d x y Nd⟨-x⟩⟨-y⟩ =
+      binary-tr
+        ( neighborhood-Metric-Ab d)
+        ( neg-neg-Metric-Ab MG x)
+        ( neg-neg-Metric-Ab MG y)
+        ( is-short-map-neg-Metric-Ab d _ _ Nd⟨-x⟩⟨-y⟩)
+
   is-isometry-neg-Metric-Ab :
     is-isometry-Metric-Space
       ( metric-space-Metric-Ab)
       ( metric-space-Metric-Ab)
       ( neg-Metric-Ab MG)
-  is-isometry-neg-Metric-Ab = pr1 (pr2 (pr2 (pr2 MG)))
+  is-isometry-neg-Metric-Ab d x y =
+    ( is-short-map-neg-Metric-Ab d x y ,
+      reflects-neighborhoods-neg-Metric-Ab d x y)
 
   isometry-neg-Metric-Ab :
     isometry-Metric-Space
@@ -253,3 +344,35 @@ module _
       ( metric-space-Metric-Ab G)
       ( modulated-uniformly-continuous-map-add-pair-Metric-Ab)
 ```
+
+### Neighborhoods of sums in metric abelian groups
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (dxx' dyy' : ℚ⁺)
+  (x x' y y' : type-Metric-Ab G)
+  where abstract
+
+  neighborhood-add-Metric-Ab :
+    neighborhood-Metric-Ab G dxx' x x' →
+    neighborhood-Metric-Ab G dyy' y y' →
+    neighborhood-Metric-Ab G
+      ( dxx' +ℚ⁺ dyy')
+      ( add-Metric-Ab G x y)
+      ( add-Metric-Ab G x' y')
+  neighborhood-add-Metric-Ab Nxx' Nyy' =
+    triangular-neighborhood-Metric-Ab G
+      ( add-Metric-Ab G x y)
+      ( add-Metric-Ab G x' y)
+      ( add-Metric-Ab G x' y')
+      ( dxx')
+      ( dyy')
+      ( forward-implication
+        ( is-isometry-add-Metric-Ab G x' dyy' y y')
+        ( Nyy'))
+      ( forward-implication
+        ( is-isometry-add-Metric-Ab' G y dxx' x x')
+        ( Nxx'))
+```
diff --git a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
index 4e2a1657a36..4ca8e619ac0 100644
--- a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
@@ -15,7 +15,9 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
 open import foundation.function-types
+open import foundation.identity-types
 open import foundation.propositions
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import order-theory.strict-preorders
@@ -64,6 +66,17 @@ leq-le-ℚ⁺ : {x y : ℚ⁺} → le-ℚ⁺ x y → leq-ℚ⁺ x y
 leq-le-ℚ⁺ {x} {y} = leq-le-ℚ {rational-ℚ⁺ x} {rational-ℚ⁺ y}
 ```
 
+### Concatenation of equality and strict inequality
+
+```agda
+module _
+  {x y z : ℚ⁺}
+  where abstract
+
+  concat-eq-le-ℚ⁺ : x ＝ y → le-ℚ⁺ y z → le-ℚ⁺ x z
+  concat-eq-le-ℚ⁺ = inv-tr (λ w → le-ℚ⁺ w z)
+```
+
 ### The strictly preordered set of positive rational numbers
 
 ```agda

From 9fcc281b33b0d4169458cf1d71bd8cae1bdd7b1f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:23:27 -0700
Subject: [PATCH 05/26] Progress

---
 src/analysis.lagda.md                         |  4 +
 ...roximations-metric-abelian-groups.lagda.md | 86 +------------------
 ...roximations-metric-abelian-groups.lagda.md | 82 ++++++++++++++++++
 3 files changed, 88 insertions(+), 84 deletions(-)
 create mode 100644 src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md

diff --git a/src/analysis.lagda.md b/src/analysis.lagda.md
index bce22009164..62e3f1ae0ca 100644
--- a/src/analysis.lagda.md
+++ b/src/analysis.lagda.md
@@ -3,13 +3,17 @@
 ```agda
 module analysis where
 
+open import analysis.addition-cauchy-approximations-metric-abelian-groups public
 open import analysis.alternation-sequences-metric-abelian-groups public
+open import analysis.cauchy-approximations-metric-abelian-groups public
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups public
 open import analysis.complete-metric-abelian-groups public
 open import analysis.convergent-series-complete-metric-abelian-groups public
 open import analysis.convergent-series-metric-abelian-groups public
 open import analysis.limits-of-sequences-metric-abelian-groups public
 open import analysis.metric-abelian-groups public
 open import analysis.metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups public
+open import analysis.negation-cauchy-approximations-metric-abelian-groups public
 open import analysis.sequences-metric-abelian-groups public
 open import analysis.series-complete-metric-abelian-groups public
 open import analysis.series-metric-abelian-groups public
diff --git a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
index 15b5c86afe3..e1b92eb14ef 100644
--- a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -25,14 +25,7 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-quotients-of-pseudometric-spaces
-open import metric-spaces.metric-spaces
 open import metric-spaces.short-maps-pseudometric-spaces
-open import metric-spaces.similarity-of-elements-pseudometric-spaces
 ```
 
 </details>
@@ -165,9 +158,7 @@ module _
 
   sim-add-const-cauchy-approximation-Metric-Ab :
     (x y : type-Metric-Ab G) →
-    sim-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Space
-        ( metric-space-Metric-Ab G))
+    sim-cauchy-pseudocompletion-Metric-Ab G
       ( add-cauchy-approximation-Metric-Ab G
         ( const-cauchy-approximation-Metric-Ab G x)
         ( const-cauchy-approximation-Metric-Ab G y))
@@ -325,80 +316,7 @@ module _
       ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab)
 ```
 
-### Negations of Cauchy approximations
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  neg-cauchy-approximation-Metric-Ab :
-    cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
-  neg-cauchy-approximation-Metric-Ab =
-    map-isometry-cauchy-approximation-Metric-Space
-      ( metric-space-Metric-Ab G)
-      ( metric-space-Metric-Ab G)
-      ( isometry-neg-Metric-Ab G)
-```
-
-### Inverse laws of Cauchy approximations
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
-  where abstract opaque
-
-  unfolding map-add-cauchy-approximation-Metric-Ab
-
-  left-inverse-law-add-cauchy-approximation-Metric-Ab :
-    add-cauchy-approximation-Metric-Ab G
-      ( neg-cauchy-approximation-Metric-Ab G ax)
-      ( ax) ＝
-    zero-cauchy-approximation-Metric-Ab G
-  left-inverse-law-add-cauchy-approximation-Metric-Ab =
-    eq-type-subtype
-      ( is-cauchy-approximation-prop-Metric-Ab G)
-      ( eq-htpy (λ _ → left-inverse-law-add-Metric-Ab G _))
-
-  right-inverse-law-add-cauchy-approximation-Metric-Ab :
-    add-cauchy-approximation-Metric-Ab G
-      ( ax)
-      ( neg-cauchy-approximation-Metric-Ab G ax) ＝
-    zero-cauchy-approximation-Metric-Ab G
-  right-inverse-law-add-cauchy-approximation-Metric-Ab =
-    eq-type-subtype
-      ( is-cauchy-approximation-prop-Metric-Ab G)
-      ( eq-htpy (λ _ → right-inverse-law-add-Metric-Ab G _))
-```
-
-### Negations of Cauchy approximations preserve similarity
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where abstract
-
-  preserves-sim-neg-cauchy-approximation-Metric-Ab :
-    (x y : cauchy-approximation-Metric-Ab G) →
-    sim-cauchy-pseudocompletion-Metric-Ab G x y →
-    sim-cauchy-pseudocompletion-Metric-Ab G
-      ( neg-cauchy-approximation-Metric-Ab G x)
-      ( neg-cauchy-approximation-Metric-Ab G y)
-  preserves-sim-neg-cauchy-approximation-Metric-Ab =
-    preserves-sim-map-isometry-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( isometry-cauchy-pseudocompletion-isometry-Metric-Space
-        ( metric-space-Metric-Ab G)
-        ( metric-space-Metric-Ab G)
-        ( isometry-neg-Metric-Ab G))
-```
-
-### Left addition preserves neighborhoods
+### Left addition is a short map in the Cauchy pseudocompletion of a metric abelian group
 
 ```agda
 module _
diff --git a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..fd96991274a
--- /dev/null
+++ b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -0,0 +1,82 @@
+# Negation of Cauchy approximations in metric abelian groups
+
+```agda
+module analysis.negation-cauchy-approximations-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.addition-cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.identity-types
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+```
+
+</details>
+
+## Idea
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  neg-cauchy-approximation-Metric-Ab :
+    cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
+  neg-cauchy-approximation-Metric-Ab =
+    map-isometry-cauchy-approximation-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-space-Metric-Ab G)
+      ( isometry-neg-Metric-Ab G)
+```
+
+## Properties
+
+### Negation is an isometry in the Cauchy pseudocompletion of metric abelian groups
+
+```agda
+
+```
+
+### Inverse laws of addition of Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  left-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( neg-cauchy-approximation-Metric-Ab G ax)
+      ( ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  left-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → left-inverse-law-add-Metric-Ab G _))
+
+  right-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( ax)
+      ( neg-cauchy-approximation-Metric-Ab G ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  right-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → right-inverse-law-add-Metric-Ab G _))
+```

From 9a21dad3b07ae2fe4ca0c48e2c89b20e0363f49f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:28:20 -0700
Subject: [PATCH 06/26] Add short maps on metric spaces

---
 src/metric-spaces.lagda.md                    |  1 +
 ...imations-short-maps-metric-spaces.lagda.md | 59 +++++++++++++++++++
 2 files changed, 60 insertions(+)
 create mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md

diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index 2e432f509dd..bd25a981e3e 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -64,6 +64,7 @@ module metric-spaces where
 open import metric-spaces.accumulation-points-subsets-located-metric-spaces public
 open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces public
 open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces public
+open import metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces public
 open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-short-maps-metric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-uniformly-continuous-maps-metric-spaces public
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
new file mode 100644
index 00000000000..0af182cc02c
--- /dev/null
+++ b/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
@@ -0,0 +1,59 @@
+# The action on Cauchy approximations of short maps on metric spaces
+
+```agda
+module metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
+open import metric-spaces.cauchy-approximations-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.short-maps-metric-spaces
+open import metric-spaces.short-maps-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+The action of [short maps](metric-spaces.short-maps-metric-spaces.md) on
+[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md) in
+[metric spaces](metric-spaces.metric-spaces.md) is itself a
+[short map](metric-spaces.short-maps-pseudometric-spaces.md) on the
+[Cauchy pseudocompletions](metric-spaces.cauchy-pseudocompletions-of-metric-spaces.md)
+of the metric spaces.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (X : Metric-Space l1 l2)
+  (Y : Metric-Space l3 l4)
+  (f : short-map-Metric-Space X Y)
+  where
+
+  short-map-cauchy-pseudocompletion-short-map-Metric-Space :
+    short-map-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space X)
+      ( cauchy-pseudocompletion-Metric-Space Y)
+  short-map-cauchy-pseudocompletion-short-map-Metric-Space =
+    short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
+
+  map-short-map-cauchy-approximation-Metric-Space :
+    cauchy-approximation-Metric-Space X →
+    cauchy-approximation-Metric-Space Y
+  map-short-map-cauchy-approximation-Metric-Space =
+    map-short-map-cauchy-approximation-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( pseudometric-Metric-Space Y)
+      ( f)
+```

From 4ec34ed2995787bf9134b45a911ff1008ef0b516 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:30:48 -0700
Subject: [PATCH 07/26] Progress

---
 ...y-approximations-metric-abelian-groups.lagda.md | 14 ++++++++++++++
 1 file changed, 14 insertions(+)

diff --git a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
index fd96991274a..3ec17db7d77 100644
--- a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -13,11 +13,13 @@ open import analysis.metric-abelian-groups
 
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import foundation.identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 
 open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
 ```
 
 </details>
@@ -46,7 +48,19 @@ module _
 ### Negation is an isometry in the Cauchy pseudocompletion of metric abelian groups
 
 ```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
 
+  abstract
+    is-isometry-neg-cauchy-pseudocompletion-Metric-Ab :
+      is-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( neg-cauchy-approximation-Metric-Ab G)
+    is-isometry-neg-cauchy-pseudocompletion-Metric-Ab =
+      {! is-isometry-map  !}
 ```
 
 ### Inverse laws of addition of Cauchy approximations

From 21322ae7e97c1026e6fdd846846c56b2b684ed2d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:32:19 -0700
Subject: [PATCH 08/26] Add proof the isometry is one

---
 ...-approximations-isometries-metric-spaces.lagda.md | 12 ++++++++++++
 1 file changed, 12 insertions(+)

diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
index 8f6e86c0afd..1937d76ea20 100644
--- a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
@@ -58,6 +58,18 @@ module _
   (f : isometry-Metric-Space X Y)
   where
 
+  abstract
+    is-isometry-cauchy-pseudocompletion-isometry-Metric-Space :
+      is-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Space X)
+        ( cauchy-pseudocompletion-Metric-Space Y)
+        ( map-isometry-cauchy-approximation-Metric-Space X Y f)
+    is-isometry-cauchy-pseudocompletion-isometry-Metric-Space =
+      is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+        ( pseudometric-Metric-Space X)
+        ( pseudometric-Metric-Space Y)
+        ( f)
+
   isometry-cauchy-pseudocompletion-isometry-Metric-Space :
     isometry-Pseudometric-Space
       ( cauchy-pseudocompletion-Metric-Space X)

From 68a99edb4b3b0f6a71087713d1b7e7d0d945f94c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:34:03 -0700
Subject: [PATCH 09/26] Progress

---
 ...hy-approximations-metric-abelian-groups.lagda.md | 13 ++++++++++++-
 1 file changed, 12 insertions(+), 1 deletion(-)

diff --git a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
index 3ec17db7d77..10a4a2b400d 100644
--- a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -60,7 +60,18 @@ module _
         ( cauchy-pseudocompletion-Metric-Ab G)
         ( neg-cauchy-approximation-Metric-Ab G)
     is-isometry-neg-cauchy-pseudocompletion-Metric-Ab =
-      {! is-isometry-map  !}
+      is-isometry-cauchy-pseudocompletion-isometry-Metric-Space
+        ( metric-space-Metric-Ab G)
+        ( metric-space-Metric-Ab G)
+        ( isometry-neg-Metric-Ab G)
+
+  isometry-neg-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+  isometry-neg-cauchy-pseudocompletion-Metric-Ab =
+    ( neg-cauchy-approximation-Metric-Ab G ,
+      is-isometry-neg-cauchy-pseudocompletion-Metric-Ab)
 ```
 
 ### Inverse laws of addition of Cauchy approximations

From 8aeeeca15780a74c6c7e71bb5decade111242ee4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:35:24 -0700
Subject: [PATCH 10/26] Progress

---
 ...n-cauchy-approximations-metric-abelian-groups.lagda.md | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
index 10a4a2b400d..6d7634281e3 100644
--- a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -9,11 +9,11 @@ module analysis.negation-cauchy-approximations-metric-abelian-groups where
 ```agda
 open import analysis.addition-cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
 
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
-open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import foundation.identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -26,6 +26,12 @@ open import metric-spaces.isometries-pseudometric-spaces
 
 ## Idea
 
+Negation of
+[Cauchy approximations](analysis.cauchy-approximations-metric-abelian-groups.md)
+in [metric abelian groups](analysis.metric-abelian-groups.md) is the inverse
+operation for
+[addition](analysis.addition-cauchy-approximations-metric-abelian-groups.md).
+
 ## Definition
 
 ```agda

From 18af96421603988a1f62e8df1931bac8d825de6f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:38:46 -0700
Subject: [PATCH 11/26] Progress

---
 ...roximations-metric-abelian-groups.lagda.md |  86 +------------
 ...roximations-metric-abelian-groups.lagda.md | 113 ++++++++++++++++++
 ...imations-isometries-metric-spaces.lagda.md |  12 ++
 ...ns-short-maps-pseudometric-spaces.lagda.md |  28 +----
 4 files changed, 128 insertions(+), 111 deletions(-)
 create mode 100644 src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md

diff --git a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
index 15b5c86afe3..e1b92eb14ef 100644
--- a/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/addition-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -25,14 +25,7 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-quotients-of-pseudometric-spaces
-open import metric-spaces.metric-spaces
 open import metric-spaces.short-maps-pseudometric-spaces
-open import metric-spaces.similarity-of-elements-pseudometric-spaces
 ```
 
 </details>
@@ -165,9 +158,7 @@ module _
 
   sim-add-const-cauchy-approximation-Metric-Ab :
     (x y : type-Metric-Ab G) →
-    sim-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Space
-        ( metric-space-Metric-Ab G))
+    sim-cauchy-pseudocompletion-Metric-Ab G
       ( add-cauchy-approximation-Metric-Ab G
         ( const-cauchy-approximation-Metric-Ab G x)
         ( const-cauchy-approximation-Metric-Ab G y))
@@ -325,80 +316,7 @@ module _
       ( sim-left-unit-law-add-cauchy-approximation-Metric-Ab)
 ```
 
-### Negations of Cauchy approximations
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  neg-cauchy-approximation-Metric-Ab :
-    cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
-  neg-cauchy-approximation-Metric-Ab =
-    map-isometry-cauchy-approximation-Metric-Space
-      ( metric-space-Metric-Ab G)
-      ( metric-space-Metric-Ab G)
-      ( isometry-neg-Metric-Ab G)
-```
-
-### Inverse laws of Cauchy approximations
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
-  where abstract opaque
-
-  unfolding map-add-cauchy-approximation-Metric-Ab
-
-  left-inverse-law-add-cauchy-approximation-Metric-Ab :
-    add-cauchy-approximation-Metric-Ab G
-      ( neg-cauchy-approximation-Metric-Ab G ax)
-      ( ax) ＝
-    zero-cauchy-approximation-Metric-Ab G
-  left-inverse-law-add-cauchy-approximation-Metric-Ab =
-    eq-type-subtype
-      ( is-cauchy-approximation-prop-Metric-Ab G)
-      ( eq-htpy (λ _ → left-inverse-law-add-Metric-Ab G _))
-
-  right-inverse-law-add-cauchy-approximation-Metric-Ab :
-    add-cauchy-approximation-Metric-Ab G
-      ( ax)
-      ( neg-cauchy-approximation-Metric-Ab G ax) ＝
-    zero-cauchy-approximation-Metric-Ab G
-  right-inverse-law-add-cauchy-approximation-Metric-Ab =
-    eq-type-subtype
-      ( is-cauchy-approximation-prop-Metric-Ab G)
-      ( eq-htpy (λ _ → right-inverse-law-add-Metric-Ab G _))
-```
-
-### Negations of Cauchy approximations preserve similarity
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where abstract
-
-  preserves-sim-neg-cauchy-approximation-Metric-Ab :
-    (x y : cauchy-approximation-Metric-Ab G) →
-    sim-cauchy-pseudocompletion-Metric-Ab G x y →
-    sim-cauchy-pseudocompletion-Metric-Ab G
-      ( neg-cauchy-approximation-Metric-Ab G x)
-      ( neg-cauchy-approximation-Metric-Ab G y)
-  preserves-sim-neg-cauchy-approximation-Metric-Ab =
-    preserves-sim-map-isometry-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( isometry-cauchy-pseudocompletion-isometry-Metric-Space
-        ( metric-space-Metric-Ab G)
-        ( metric-space-Metric-Ab G)
-        ( isometry-neg-Metric-Ab G))
-```
-
-### Left addition preserves neighborhoods
+### Left addition is a short map in the Cauchy pseudocompletion of a metric abelian group
 
 ```agda
 module _
diff --git a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..6d7634281e3
--- /dev/null
+++ b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -0,0 +1,113 @@
+# Negation of Cauchy approximations in metric abelian groups
+
+```agda
+module analysis.negation-cauchy-approximations-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.addition-cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.identity-types
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
+```
+
+</details>
+
+## Idea
+
+Negation of
+[Cauchy approximations](analysis.cauchy-approximations-metric-abelian-groups.md)
+in [metric abelian groups](analysis.metric-abelian-groups.md) is the inverse
+operation for
+[addition](analysis.addition-cauchy-approximations-metric-abelian-groups.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  neg-cauchy-approximation-Metric-Ab :
+    cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
+  neg-cauchy-approximation-Metric-Ab =
+    map-isometry-cauchy-approximation-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-space-Metric-Ab G)
+      ( isometry-neg-Metric-Ab G)
+```
+
+## Properties
+
+### Negation is an isometry in the Cauchy pseudocompletion of metric abelian groups
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  abstract
+    is-isometry-neg-cauchy-pseudocompletion-Metric-Ab :
+      is-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( neg-cauchy-approximation-Metric-Ab G)
+    is-isometry-neg-cauchy-pseudocompletion-Metric-Ab =
+      is-isometry-cauchy-pseudocompletion-isometry-Metric-Space
+        ( metric-space-Metric-Ab G)
+        ( metric-space-Metric-Ab G)
+        ( isometry-neg-Metric-Ab G)
+
+  isometry-neg-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+  isometry-neg-cauchy-pseudocompletion-Metric-Ab =
+    ( neg-cauchy-approximation-Metric-Ab G ,
+      is-isometry-neg-cauchy-pseudocompletion-Metric-Ab)
+```
+
+### Inverse laws of addition of Cauchy approximations
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  (ax@(x , is-approx-x) : cauchy-approximation-Metric-Ab G)
+  where abstract opaque
+
+  unfolding map-add-cauchy-approximation-Metric-Ab
+
+  left-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( neg-cauchy-approximation-Metric-Ab G ax)
+      ( ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  left-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → left-inverse-law-add-Metric-Ab G _))
+
+  right-inverse-law-add-cauchy-approximation-Metric-Ab :
+    add-cauchy-approximation-Metric-Ab G
+      ( ax)
+      ( neg-cauchy-approximation-Metric-Ab G ax) ＝
+    zero-cauchy-approximation-Metric-Ab G
+  right-inverse-law-add-cauchy-approximation-Metric-Ab =
+    eq-type-subtype
+      ( is-cauchy-approximation-prop-Metric-Ab G)
+      ( eq-htpy (λ _ → right-inverse-law-add-Metric-Ab G _))
+```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
index 8f6e86c0afd..1937d76ea20 100644
--- a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
+++ b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
@@ -58,6 +58,18 @@ module _
   (f : isometry-Metric-Space X Y)
   where
 
+  abstract
+    is-isometry-cauchy-pseudocompletion-isometry-Metric-Space :
+      is-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Space X)
+        ( cauchy-pseudocompletion-Metric-Space Y)
+        ( map-isometry-cauchy-approximation-Metric-Space X Y f)
+    is-isometry-cauchy-pseudocompletion-isometry-Metric-Space =
+      is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
+        ( pseudometric-Metric-Space X)
+        ( pseudometric-Metric-Space Y)
+        ( f)
+
   isometry-cauchy-pseudocompletion-isometry-Metric-Space :
     isometry-Pseudometric-Space
       ( cauchy-pseudocompletion-Metric-Space X)
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
index 368391bef67..281e54b3bd9 100644
--- a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
@@ -11,14 +11,12 @@ open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.dependent-pair-types
-open import foundation.functoriality-set-quotients
 open import foundation.universe-levels
 
 open import metric-spaces.cauchy-approximations-pseudometric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
 open import metric-spaces.pseudometric-spaces
 open import metric-spaces.short-maps-pseudometric-spaces
-open import metric-spaces.similarity-of-elements-pseudometric-spaces
 ```
 
 </details>
@@ -64,7 +62,7 @@ module _
 
 ## Properties
 
-### Short maps preserve neighborhoods in the Cauchy pseudocompletion of a pseudometric space
+### The induced action of a short map on Cauchy approximations is short in the Cauchy pseudocompletion of a pseudometric space
 
 ```agda
 module _
@@ -92,27 +90,3 @@ module _
     ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f ,
       is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space)
 ```
-
-### Short maps preserve similarity in the Cauchy pseudocompletion of a pseudometric space
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Pseudometric-Space l1 l2)
-  (Y : Pseudometric-Space l3 l4)
-  (f : short-map-Pseudometric-Space X Y)
-  where abstract
-
-  preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space :
-    preserves-sim-equivalence-relation
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space X)
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Pseudometric-Space Y)
-      ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f)
-  preserves-sim-short-map-cauchy-pseudocompletion-Pseudometric-Space {x} {y} =
-    preserves-sim-map-short-map-Pseudometric-Space
-      ( cauchy-pseudocompletion-Pseudometric-Space X)
-      ( cauchy-pseudocompletion-Pseudometric-Space Y)
-      ( short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space X Y f)
-      ( x)
-      ( y)
-```

From a3f9fd19155b5f9cb6bbd58fea48fa2b545abdc1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:42:26 -0700
Subject: [PATCH 12/26] Update

---
 src/analysis.lagda.md                                |  4 ++--
 ...pseudocompletions-metric-abelian-groups.lagda.md} | 12 +++++++++---
 ...pseudocompletions-metric-abelian-groups.lagda.md} |  4 ++--
 3 files changed, 13 insertions(+), 7 deletions(-)
 rename src/analysis/{abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md => abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md} (97%)
 rename src/analysis/{metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md => metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md} (97%)

diff --git a/src/analysis.lagda.md b/src/analysis.lagda.md
index 74269fbccad..5c8faa60fb0 100644
--- a/src/analysis.lagda.md
+++ b/src/analysis.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 module analysis where
 
-open import analysis.abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups public
+open import analysis.abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups public
 open import analysis.addition-cauchy-approximations-metric-abelian-groups public
 open import analysis.alternation-sequences-metric-abelian-groups public
 open import analysis.cauchy-approximations-metric-abelian-groups public
@@ -12,8 +12,8 @@ open import analysis.complete-metric-abelian-groups public
 open import analysis.convergent-series-complete-metric-abelian-groups public
 open import analysis.convergent-series-metric-abelian-groups public
 open import analysis.limits-of-sequences-metric-abelian-groups public
-open import analysis.metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups public
 open import analysis.metric-abelian-groups public
+open import analysis.metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups public
 open import analysis.metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups public
 open import analysis.negation-cauchy-approximations-metric-abelian-groups public
 open import analysis.sequences-metric-abelian-groups public
diff --git a/src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
similarity index 97%
rename from src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
rename to src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index db37e3e37b8..56100ab7564 100644
--- a/src/analysis/abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
+++ b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -1,9 +1,9 @@
-# The abelian group of the metric quotient of Cauchy pseudocompletions of metric abelian groups
+# The abelian groups of the metric quotients of Cauchy pseudocompletions of metric abelian groups
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module analysis.abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups where
+module analysis.abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups where
 ```
 
 <details><summary>Imports</summary>
@@ -13,6 +13,7 @@ open import analysis.addition-cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
+open import analysis.negation-cauchy-approximations-metric-abelian-groups
 
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -428,7 +429,12 @@ module _
       ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
   hom-neg-cauchy-pseudocompletion-Metric-Ab =
     ( neg-cauchy-approximation-Metric-Ab G ,
-      preserves-sim-neg-cauchy-approximation-Metric-Ab G _ _)
+      preserves-sim-map-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( isometry-neg-cauchy-pseudocompletion-Metric-Ab G)
+        ( _)
+        ( _))
 
   neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
     type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G →
diff --git a/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
similarity index 97%
rename from src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
rename to src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index c8d4b7bca3b..8a32a696049 100644
--- a/src/analysis/metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -3,13 +3,13 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module analysis.metric-abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups where
+module analysis.metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups where
 ```
 
 <details><summary>Imports</summary>
 
 ```agda
-open import analysis.abelian-group-metric-quotient-cauchy-pseudocompletion-metric-abelian-groups
+open import analysis.abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.addition-cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups

From 7725d45d34d09b870f8a59e0bdd464bfae74eb53 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:43:24 -0700
Subject: [PATCH 13/26] Fix codespell

---
 config/codespell-ignore.txt | 1 +
 1 file changed, 1 insertion(+)

diff --git a/config/codespell-ignore.txt b/config/codespell-ignore.txt
index 79a8f895b86..e060a3a5f41 100644
--- a/config/codespell-ignore.txt
+++ b/config/codespell-ignore.txt
@@ -1,3 +1,4 @@
+Nd
 Tim
 blacklist
 couldn

From a2e6047f42a156f16209d5634effa94a28b9283e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:43:56 -0700
Subject: [PATCH 14/26] Fix link

---
 .../cauchy-pseudocompletions-metric-abelian-groups.lagda.md     | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 63b1685f2e2..ec533984187 100644
--- a/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -29,7 +29,7 @@ open import metric-spaces.similarity-of-elements-pseudometric-spaces
 
 The
 {{#concept "Cauchy pseudocompletion" Disambiguation="of a metric abelian group" Agda=cauchy-pseudocompletion-Metric-Ab}}
-of a [metric abelian group](group-theory.metric-abelian-groups.md) is the
+of a [metric abelian group](analysis.metric-abelian-groups.md) is the
 [Cauchy pseudocompletion](metric-spaces.cauchy-pseudocompletions-of-metric-spaces.md)
 of the underlying [metric space](metric-spaces.metric-spaces.md).
 

From a89071f1869db2d3e410b73839e07f25d0c27945 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 14:58:07 -0700
Subject: [PATCH 15/26] Progress

---
 ...completions-metric-abelian-groups.lagda.md |  4 +-
 ...completions-metric-abelian-groups.lagda.md | 53 +++++++++++++++++++
 .../homomorphisms-abelian-groups.lagda.md     |  5 +-
 3 files changed, 58 insertions(+), 4 deletions(-)

diff --git a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 56100ab7564..538d1cbe5c2 100644
--- a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -216,13 +216,13 @@ module _
       ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
       ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab G)
 
-  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+  add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
     (x y : type-Metric-Ab G) →
     add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
       ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G x)
       ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G y) ＝
     in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G (add-Metric-Ab G x y)
-  preserves-add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x y =
+  add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x y =
     add-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
       ( const-cauchy-approximation-Metric-Ab G x)
       ( const-cauchy-approximation-Metric-Ab G y) ∙
diff --git a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 8a32a696049..75d663a13e0 100644
--- a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -26,12 +26,14 @@ open import foundation.set-quotients
 open import foundation.universe-levels
 
 open import group-theory.abelian-groups
+open import group-theory.homomorphisms-abelian-groups
 
 open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.short-maps-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.short-maps-pseudometric-spaces
 open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 ```
@@ -229,3 +231,54 @@ module _
       is-short-map-neg-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
       is-short-map-left-add-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
 ```
+
+## Properties
+
+### The embedding of the metric abelian group into the metric abelian group of the metric quotient of its Cauchy pseudocompletion is an isometry
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( pseudometric-space-Metric-Ab
+        ( metric-ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+  isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    isometry-unit-metric-quotient-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+
+  isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+  isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    comp-isometry-Pseudometric-Space
+      ( pseudometric-space-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( pseudometric-Metric-Space
+        ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
+      ( isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab )
+      ( isometry-unit-cauchy-pseudocompletion-Metric-Space
+        ( metric-space-Metric-Ab G))
+```
+
+### The embedding of the metric abelian group into the metric abelian group of the metric quotient of its Cauchy pseudocompletion is an Abelian group homomorphism
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  hom-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    hom-Ab
+      ( ab-Metric-Ab G)
+      ( ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+  hom-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G ,
+      inv (add-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab G _ _))
+```
diff --git a/src/group-theory/homomorphisms-abelian-groups.lagda.md b/src/group-theory/homomorphisms-abelian-groups.lagda.md
index 43fa154acc8..5ad470912c9 100644
--- a/src/group-theory/homomorphisms-abelian-groups.lagda.md
+++ b/src/group-theory/homomorphisms-abelian-groups.lagda.md
@@ -28,8 +28,9 @@ open import group-theory.homomorphisms-semigroups
 
 ## Idea
 
-Homomorphisms between abelian groups are just homomorphisms between their
-underlying groups.
+Homomorphisms between [abelian groups](group-theory.abelian-groups.md) are just
+[homomorphisms](group-theory.homomorphisms-groups.md) between their underlying
+[groups](group-theory.groups.md).
 
 ## Definition
 

From 3e76740955d8525a39221fef6a71756b604135f5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 15:15:18 -0700
Subject: [PATCH 16/26] Progress

---
 src/analysis.lagda.md                         |  1 +
 ...completions-metric-abelian-groups.lagda.md | 45 +-----------
 ...completions-metric-abelian-groups.lagda.md | 69 +++++++++++++++++++
 3 files changed, 73 insertions(+), 42 deletions(-)
 create mode 100644 src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md

diff --git a/src/analysis.lagda.md b/src/analysis.lagda.md
index 5c8faa60fb0..74ccc276304 100644
--- a/src/analysis.lagda.md
+++ b/src/analysis.lagda.md
@@ -15,6 +15,7 @@ open import analysis.limits-of-sequences-metric-abelian-groups public
 open import analysis.metric-abelian-groups public
 open import analysis.metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups public
 open import analysis.metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups public
+open import analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups public
 open import analysis.negation-cauchy-approximations-metric-abelian-groups public
 open import analysis.sequences-metric-abelian-groups public
 open import analysis.series-complete-metric-abelian-groups public
diff --git a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 538d1cbe5c2..c9e6f29fffa 100644
--- a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -13,6 +13,7 @@ open import analysis.addition-cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
+open import analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.negation-cauchy-approximations-metric-abelian-groups
 
 open import elementary-number-theory.addition-positive-rational-numbers
@@ -58,51 +59,11 @@ open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 The [metric quotient](metric-spaces.metric-quotients-of-pseudometric-spaces.md)
 of the
 [Cauchy pseudocompletion](analysis.cauchy-pseudocompletions-metric-abelian-groups.md)
-of a [metric abelian group](analysis.metric-abelian-groups.md) itself forms a
-metric abelian group.
-
-This construction is precisely analogous to the definition of the Cauchy real
-numbers and their definition of addition.
+of a [metric abelian group](analysis.metric-abelian-groups.md) forms an
+[abelian group](group-theory.abelian-groups.md).
 
 ## Definition
 
-### The metric space of the metric quotient
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    Metric-Space (l1 ⊔ l2) (l1 ⊔ l2)
-  metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    metric-quotient-Pseudometric-Space (cauchy-pseudocompletion-Metric-Ab G)
-
-  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Set (l1 ⊔ l2)
-  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    set-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
-
-  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab : UU (l1 ⊔ l2)
-  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    type-Set set-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-
-  neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    Rational-Neighborhood-Relation
-      ( l1 ⊔ l2)
-      ( type-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-  neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    neighborhood-prop-Metric-Space
-      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-
-  neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    ℚ⁺ →
-    Relation (l1 ⊔ l2) type-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-  neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    neighborhood-Metric-Space
-      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-```
-
 ### Addition in the metric quotient
 
 ```agda
diff --git a/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
new file mode 100644
index 00000000000..60c9bd06928
--- /dev/null
+++ b/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -0,0 +1,69 @@
+# Metric quotients of Cauchy pseudocompletions of metric abelian groups
+
+```agda
+module analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-abelian-groups
+
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.binary-relations
+open import foundation.sets
+open import foundation.universe-levels
+
+open import metric-spaces.metric-quotients-of-pseudometric-spaces
+open import metric-spaces.metric-spaces
+open import metric-spaces.rational-neighborhood-relations
+```
+
+</details>
+
+## Idea
+
+The [metric quotient](metric-spaces.metric-quotients-of-pseudometric-spaces.md)
+of the
+[Cauchy pseudocompletion](analysis.cauchy-pseudocompletions-metric-abelian-groups.md)
+of a [metric abelian group](analysis.metric-abelian-groups.md) forms a
+[metric space](metric-spaces.metric-spaces.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Metric-Space (l1 ⊔ l2) (l1 ⊔ l2)
+  metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    metric-quotient-Pseudometric-Space (cauchy-pseudocompletion-Metric-Ab G)
+
+  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Set (l1 ⊔ l2)
+  set-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    set-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab : UU (l1 ⊔ l2)
+  type-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    type-Set set-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
+  neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Rational-Neighborhood-Relation
+      ( l1 ⊔ l2)
+      ( type-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+  neighborhood-prop-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-prop-Metric-Space
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+
+  neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    ℚ⁺ →
+    Relation (l1 ⊔ l2) type-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+  neighborhood-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    neighborhood-Metric-Space
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+```

From a75d5dc43973a29d1cff1b2bfc68ead1fedf81d5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 15:15:46 -0700
Subject: [PATCH 17/26] Progress

---
 ...ients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 75d663a13e0..1d400421c65 100644
--- a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -10,6 +10,7 @@ module analysis.metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-
 
 ```agda
 open import analysis.abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
+open import analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.addition-cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups

From d9fc3e584fb339f54188179a880dca51f9fea205 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 15:15:57 -0700
Subject: [PATCH 18/26] pre-commit

---
 ...-cauchy-pseudocompletions-metric-abelian-groups.lagda.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 1d400421c65..b3805f4b9c3 100644
--- a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -10,10 +10,10 @@ module analysis.metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-
 
 ```agda
 open import analysis.abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
-open import analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.addition-cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
+open import analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 
 open import elementary-number-theory.positive-rational-numbers
 
@@ -30,11 +30,11 @@ open import group-theory.abelian-groups
 open import group-theory.homomorphisms-abelian-groups
 
 open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.short-maps-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.short-maps-pseudometric-spaces
 open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 ```
@@ -262,7 +262,7 @@ module _
       ( cauchy-pseudocompletion-Metric-Ab G)
       ( pseudometric-Metric-Space
         ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
-      ( isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab )
+      ( isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
       ( isometry-unit-cauchy-pseudocompletion-Metric-Space
         ( metric-space-Metric-Ab G))
 ```

From f34a2038fba251d6567df210b28601fd9a704f78 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 15:42:33 -0700
Subject: [PATCH 19/26] Move isometries into the metric quotient file

---
 ...completions-metric-abelian-groups.lagda.md | 54 -------------
 ...completions-metric-abelian-groups.lagda.md | 80 +++++++++++++++++++
 2 files changed, 80 insertions(+), 54 deletions(-)

diff --git a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index c9e6f29fffa..0c451f6dd99 100644
--- a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -94,60 +94,6 @@ module _
       ( binary-hom-add-cauchy-pseudocompletion-Metric-Ab)
 ```
 
-### The embedding of elements of a metric abelian group in the metric quotient of its Cauchy pseudocompletion
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    cauchy-approximation-Metric-Ab G →
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    quotient-map
-      ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-
-  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-Metric-Ab G → type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
-    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-      ( const-cauchy-approximation-Metric-Ab G x)
-
-  abstract
-    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-      is-injective in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-    is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-      {x} {y} inx=iny =
-      eq-sim-const-cauchy-approximation-cauchy-pseudocompletion-Metric-Ab
-        ( G)
-        ( x)
-        ( y)
-        ( apply-effectiveness-quotient-map
-          ( equivalence-relation-sim-cauchy-pseudocompletion-Metric-Ab G)
-          ( inx=iny))
-
-    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-      is-emb in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-      is-emb-is-injective
-        ( is-set-type-Metric-Space
-          ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
-        is-injective-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
-
-  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-Metric-Ab G ↪ type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
-      is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-
-  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
-  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab (zero-Metric-Ab G)
-```
-
 ## Properties
 
 ### The embedding in the metric quotient of the Cauchy pseudocompletion preserves addition
diff --git a/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 60c9bd06928..e5a1f75e820 100644
--- a/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -1,6 +1,8 @@
 # Metric quotients of Cauchy pseudocompletions of metric abelian groups
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups where
 ```
 
@@ -9,15 +11,23 @@ module analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 ```agda
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
+open import analysis.cauchy-approximations-metric-abelian-groups
+open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 
 open import elementary-number-theory.positive-rational-numbers
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 
+open import foundation.embeddings
 open import foundation.binary-relations
+open import foundation.dependent-pair-types
 open import foundation.sets
 open import foundation.universe-levels
+open import metric-spaces.pseudometric-spaces
 
+open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-quotients-of-pseudometric-spaces
 open import metric-spaces.metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.rational-neighborhood-relations
 ```
 
@@ -44,6 +54,11 @@ module _
   metric-quotient-cauchy-pseudocompletion-Metric-Ab =
     metric-quotient-Pseudometric-Space (cauchy-pseudocompletion-Metric-Ab G)
 
+  pseudometric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    Pseudometric-Space (l1 ⊔ l2) (l1 ⊔ l2)
+  pseudometric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    pseudometric-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
+
   set-metric-quotient-cauchy-pseudocompletion-Metric-Ab : Set (l1 ⊔ l2)
   set-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
     set-Metric-Space metric-quotient-cauchy-pseudocompletion-Metric-Ab
@@ -67,3 +82,68 @@ module _
     neighborhood-Metric-Space
       ( metric-quotient-cauchy-pseudocompletion-Metric-Ab)
 ```
+
+## Properties
+
+### The embedding of elements of a metric abelian group in the metric quotient of its Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level}
+  (G : Metric-Ab l1 l2)
+  where
+
+  isometry-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( pseudometric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+  isometry-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    isometry-unit-metric-quotient-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+
+  isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+  isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    comp-isometry-Pseudometric-Space
+      ( pseudometric-space-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( pseudometric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+      ( isometry-in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+      ( isometry-unit-cauchy-pseudocompletion-Metric-Space
+        ( metric-space-Metric-Ab G))
+
+  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    cauchy-approximation-Metric-Ab G →
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    map-unit-metric-quotient-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+
+  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-Metric-Ab G → type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  in-metric-quotient-cauchy-pseudocompletion-Metric-Ab x =
+    in-approximation-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+      ( const-cauchy-approximation-Metric-Ab G x)
+
+  abstract
+    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+      is-emb in-metric-quotient-cauchy-pseudocompletion-Metric-Ab
+    is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+      is-emb-map-isometry-Metric-Space
+        ( metric-space-Metric-Ab G)
+        ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
+        ( isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+
+  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-Metric-Ab G ↪ type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  emb-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    ( in-metric-quotient-cauchy-pseudocompletion-Metric-Ab ,
+      is-emb-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
+
+  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
+    type-metric-quotient-cauchy-pseudocompletion-Metric-Ab G
+  zero-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
+    in-metric-quotient-cauchy-pseudocompletion-Metric-Ab (zero-Metric-Ab G)
+```

From ad0b7d8f465548adf05e87bc7f5aa57fca179994 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 15:45:32 -0700
Subject: [PATCH 20/26] Fix imports

---
 ...seudocompletions-metric-abelian-groups.lagda.md | 14 --------------
 ...seudocompletions-metric-abelian-groups.lagda.md | 12 ++++++------
 2 files changed, 6 insertions(+), 20 deletions(-)

diff --git a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 0c451f6dd99..a5ae7f941e6 100644
--- a/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -22,15 +22,9 @@ open import elementary-number-theory.positive-rational-numbers
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-functoriality-set-quotients
-open import foundation.binary-relations
-open import foundation.binary-transport
 open import foundation.dependent-pair-types
-open import foundation.embeddings
-open import foundation.equivalence-relations
 open import foundation.functoriality-set-quotients
 open import foundation.identity-types
-open import foundation.injective-maps
-open import foundation.propositional-truncations
 open import foundation.set-quotients
 open import foundation.sets
 open import foundation.transport-along-identifications
@@ -41,15 +35,7 @@ open import group-theory.groups
 open import group-theory.monoids
 open import group-theory.semigroups
 
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-quotients-of-pseudometric-spaces
-open import metric-spaces.metric-spaces
-open import metric-spaces.pseudometric-spaces
-open import metric-spaces.rational-neighborhood-relations
 open import metric-spaces.similarity-of-elements-pseudometric-spaces
-open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 ```
 
 </details>
diff --git a/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index e5a1f75e820..99750ce71e2 100644
--- a/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -9,26 +9,26 @@ module analysis.metric-quotients-cauchy-pseudocompletions-metric-abelian-groups
 <details><summary>Imports</summary>
 
 ```agda
+open import analysis.cauchy-approximations-metric-abelian-groups
 open import analysis.cauchy-pseudocompletions-metric-abelian-groups
 open import analysis.metric-abelian-groups
-open import analysis.cauchy-approximations-metric-abelian-groups
-open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 
 open import elementary-number-theory.positive-rational-numbers
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 
-open import foundation.embeddings
 open import foundation.binary-relations
 open import foundation.dependent-pair-types
+open import foundation.embeddings
 open import foundation.sets
 open import foundation.universe-levels
-open import metric-spaces.pseudometric-spaces
 
+open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.isometries-metric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-quotients-of-pseudometric-spaces
 open import metric-spaces.metric-spaces
-open import metric-spaces.isometries-pseudometric-spaces
+open import metric-spaces.pseudometric-spaces
 open import metric-spaces.rational-neighborhood-relations
+open import metric-spaces.unit-map-metric-quotients-of-pseudometric-spaces
 ```
 
 </details>

From 4475b52340083ad9e3fb5ce947eb314cdc953df8 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 15:53:43 -0700
Subject: [PATCH 21/26] Fix

---
 ...completions-metric-abelian-groups.lagda.md | 32 -------------------
 1 file changed, 32 deletions(-)

diff --git a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index b3805f4b9c3..30c1bf7a15a 100644
--- a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -235,38 +235,6 @@ module _
 
 ## Properties
 
-### The embedding of the metric abelian group into the metric abelian group of the metric quotient of its Cauchy pseudocompletion is an isometry
-
-```agda
-module _
-  {l1 l2 : Level}
-  (G : Metric-Ab l1 l2)
-  where
-
-  isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    isometry-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( pseudometric-space-Metric-Ab
-        ( metric-ab-metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
-  isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    isometry-unit-metric-quotient-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Ab G)
-
-  isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab :
-    isometry-Metric-Space
-      ( metric-space-Metric-Ab G)
-      ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G)
-  isometry-in-metric-quotient-cauchy-pseudocompletion-Metric-Ab =
-    comp-isometry-Pseudometric-Space
-      ( pseudometric-space-Metric-Ab G)
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( pseudometric-Metric-Space
-        ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
-      ( isometry-in-approx-metric-quotient-cauchy-pseudocompletion-Metric-Ab)
-      ( isometry-unit-cauchy-pseudocompletion-Metric-Space
-        ( metric-space-Metric-Ab G))
-```
-
 ### The embedding of the metric abelian group into the metric abelian group of the metric quotient of its Cauchy pseudocompletion is an Abelian group homomorphism
 
 ```agda

From 18294cf3a3341b826715f780c358ffd842ff44bb Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 13 Jun 2026 18:51:46 -0700
Subject: [PATCH 22/26] Pull in changes from further along

---
 ...ompletions-of-pseudometric-spaces.lagda.md |   2 +-
 ...uchy-approximations-metric-spaces.lagda.md | 134 ++++++++++++++++++
 2 files changed, 135 insertions(+), 1 deletion(-)

diff --git a/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md b/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
index 205625ab962..9e9c8c5750e 100644
--- a/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-pseudocompletions-of-pseudometric-spaces.lagda.md
@@ -433,7 +433,7 @@ module _
   {l1 l2 : Level} (M : Pseudometric-Space l1 l2)
   (u v : cauchy-approximation-Pseudometric-Space M)
   (x : type-Pseudometric-Space M)
-  where
+  where abstract
 
   has-same-limit-sim-cauchy-approximation-Pseudometric-Space :
     sim-Pseudometric-Space
diff --git a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
index 6a2459d92bd..524cf3c7a21 100644
--- a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
@@ -17,6 +17,8 @@ open import foundation.universe-levels
 
 open import metric-spaces.cauchy-approximations-metric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
+open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.limits-of-cauchy-approximations-pseudometric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.short-maps-metric-spaces
@@ -183,6 +185,66 @@ module _
       ( λ d → H d α β)
 ```
 
+### Cauchy approximations with the same limit are similar in the Cauchy pseudocompletion
+
+```agda
+module _
+  {l1 l2 : Level} (M : Metric-Space l1 l2)
+  (u v : cauchy-approximation-Metric-Space M)
+  (x : type-Metric-Space M)
+  (is-limit-u-x : is-limit-cauchy-approximation-Metric-Space M u x)
+  (is-limit-v-x : is-limit-cauchy-approximation-Metric-Space M v x)
+  where abstract
+
+  sim-is-limit-cauchy-approximation-Metric-Space :
+    sim-cauchy-pseudocompletion-Metric-Space M u v
+  sim-is-limit-cauchy-approximation-Metric-Space =
+    transitive-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space M)
+      ( u)
+      ( const-cauchy-approximation-Metric-Space M x)
+      ( v)
+      ( symmetric-sim-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Space M)
+        ( v)
+        ( const-cauchy-approximation-Metric-Space M x)
+        ( sim-const-is-limit-cauchy-approximation-Metric-Space M
+          ( v)
+          ( x)
+          ( is-limit-v-x)))
+      ( sim-const-is-limit-cauchy-approximation-Metric-Space M u x is-limit-u-x)
+```
+
+### If two Cauchy approximations are similar and have limits, the limits are equal
+
+```agda
+module _
+  {l1 l2 : Level}
+  (M : Metric-Space l1 l2)
+  (u v : cauchy-approximation-Metric-Space M)
+  (u~v : sim-cauchy-pseudocompletion-Metric-Space M u v)
+  (x y : type-Metric-Space M)
+  (is-lim-u-x : is-limit-cauchy-approximation-Metric-Space M u x)
+  (is-lim-v-y : is-limit-cauchy-approximation-Metric-Space M v y)
+  where abstract
+
+  eq-limit-sim-cauchy-pseudocompletion-Metric-Space : x ＝ y
+  eq-limit-sim-cauchy-pseudocompletion-Metric-Space =
+    all-eq-is-limit-cauchy-approximation-Metric-Space
+      ( M)
+      ( v)
+      ( x)
+      ( y)
+      ( has-same-limit-sim-cauchy-approximation-Pseudometric-Space
+        ( pseudometric-Metric-Space M)
+        ( u)
+        ( v)
+        ( x)
+        ( u~v)
+        ( is-lim-u-x))
+      ( is-lim-v-y)
+```
+
 ### Homotopic Cauchy approximations have the same limits
 
 ```agda
@@ -205,6 +267,78 @@ module _
       ( f~g)
 ```
 
+### If two Cauchy approximations have limits, they are in a `d`-neighborhood in the Cauchy pseudocompletion if and only if their limits are in a `d`-neighborhood
+
+```agda
+module _
+  {l1 l2 : Level}
+  (X : Metric-Space l1 l2)
+  (d : ℚ⁺)
+  (f g : cauchy-approximation-Metric-Space X)
+  (x y : type-Metric-Space X)
+  (is-lim-f-x : is-limit-cauchy-approximation-Metric-Space X f x)
+  (is-lim-g-y : is-limit-cauchy-approximation-Metric-Space X g y)
+  where abstract
+
+  preserves-neighborhoods-limits-cauchy-approximation-Metric-Space :
+    neighborhood-cauchy-pseudocompletion-Metric-Space X d f g →
+    neighborhood-Metric-Space X d x y
+  preserves-neighborhoods-limits-cauchy-approximation-Metric-Space Ndfg =
+    reflects-neighborhoods-map-isometry-Pseudometric-Space
+      ( pseudometric-Metric-Space X)
+      ( cauchy-pseudocompletion-Metric-Space X)
+      ( isometry-unit-cauchy-pseudocompletion-Metric-Space X)
+      ( d)
+      ( x)
+      ( y)
+      ( preserves-neighborhoods-sim-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Space X)
+        { x = f}
+        { x' = const-cauchy-approximation-Metric-Space X x}
+        { y = g}
+        { y' = const-cauchy-approximation-Metric-Space X y}
+        ( sim-const-is-limit-cauchy-approximation-Metric-Space X f x is-lim-f-x)
+        ( sim-const-is-limit-cauchy-approximation-Metric-Space X g y is-lim-g-y)
+        ( d)
+        ( Ndfg))
+
+  reflects-neighborhoods-limits-cauchy-approximation-Metric-Space :
+    neighborhood-Metric-Space X d x y →
+    neighborhood-cauchy-pseudocompletion-Metric-Space X d f g
+  reflects-neighborhoods-limits-cauchy-approximation-Metric-Space Ndxy =
+    preserves-neighborhoods-sim-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Space X)
+      { x = const-cauchy-approximation-Metric-Space X x}
+      { x' = f}
+      { y = const-cauchy-approximation-Metric-Space X y}
+      { y' = g}
+      ( symmetric-sim-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Space X)
+        ( f)
+        ( const-cauchy-approximation-Metric-Space X x)
+        ( sim-const-is-limit-cauchy-approximation-Metric-Space X
+          ( f)
+          ( x)
+          ( is-lim-f-x)))
+      ( symmetric-sim-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Space X)
+        ( g)
+        ( const-cauchy-approximation-Metric-Space X y)
+        ( sim-const-is-limit-cauchy-approximation-Metric-Space X
+          ( g)
+          ( y)
+          ( is-lim-g-y)))
+      ( d)
+      ( preserves-neighborhoods-map-isometry-Pseudometric-Space
+        ( pseudometric-Metric-Space X)
+        ( cauchy-pseudocompletion-Metric-Space X)
+        ( isometry-unit-cauchy-pseudocompletion-Metric-Space X)
+        ( d)
+        ( x)
+        ( y)
+        ( Ndxy))
+```
+
 ## See also
 
 - [Convergent cauchy approximations](metric-spaces.convergent-cauchy-approximations-metric-spaces.md)

From e3b2cb5edb27561034b7b977e73f5b217ec19061 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 14 Jun 2026 11:13:01 -0700
Subject: [PATCH 23/26] Remove redundant logic

---
 src/metric-spaces.lagda.md                    |   4 -
 ...imations-isometries-metric-spaces.lagda.md |  82 ------------
 ...ns-isometries-pseudometric-spaces.lagda.md | 102 --------------
 ...imations-short-maps-metric-spaces.lagda.md |  59 ---------
 ...ns-short-maps-pseudometric-spaces.lagda.md |  92 -------------
 ...uchy-approximations-metric-spaces.lagda.md | 124 +++++++++++-------
 .../metric-space-of-rational-numbers.lagda.md |  31 ++++-
 ...y-of-elements-pseudometric-spaces.lagda.md |  22 ++++
 8 files changed, 120 insertions(+), 396 deletions(-)
 delete mode 100644 src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
 delete mode 100644 src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
 delete mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
 delete mode 100644 src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md

diff --git a/src/metric-spaces.lagda.md b/src/metric-spaces.lagda.md
index bd25a981e3e..89f28adc578 100644
--- a/src/metric-spaces.lagda.md
+++ b/src/metric-spaces.lagda.md
@@ -62,10 +62,6 @@ metric space, `N d₂ x y` [or](foundation.disjunction.md)
 module metric-spaces where
 
 open import metric-spaces.accumulation-points-subsets-located-metric-spaces public
-open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces public
-open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces public
-open import metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces public
-open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-short-maps-metric-spaces public
 open import metric-spaces.action-on-cauchy-sequences-uniformly-continuous-maps-metric-spaces public
 open import metric-spaces.action-on-convergent-sequences-modulated-uniformly-continuous-maps-metric-spaces public
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
deleted file mode 100644
index 1937d76ea20..00000000000
--- a/src/metric-spaces/action-on-cauchy-approximations-isometries-metric-spaces.lagda.md
+++ /dev/null
@@ -1,82 +0,0 @@
-# The action on Cauchy approximations of isometries on metric spaces
-
-```agda
-module metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import foundation.functoriality-set-quotients
-open import foundation.universe-levels
-
-open import metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.isometries-pseudometric-spaces
-open import metric-spaces.metric-spaces
-```
-
-</details>
-
-## Idea
-
-The action of [isometries](metric-spaces.isometries-metric-spaces.md) on
-[metric spaces](metric-spaces.metric-spaces.md) preserves
-[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md).
-
-## Definition
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Metric-Space l1 l2)
-  (Y : Metric-Space l3 l4)
-  (f : isometry-Metric-Space X Y)
-  where
-
-  map-isometry-cauchy-approximation-Metric-Space :
-    cauchy-approximation-Metric-Space X →
-    cauchy-approximation-Metric-Space Y
-  map-isometry-cauchy-approximation-Metric-Space =
-    map-isometry-cauchy-approximation-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( pseudometric-Metric-Space Y)
-      ( f)
-```
-
-## Properties
-
-### An isometry on metric spaces induces an isometry on the Cauchy pseudocompletions of the metric spaces
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Metric-Space l1 l2)
-  (Y : Metric-Space l3 l4)
-  (f : isometry-Metric-Space X Y)
-  where
-
-  abstract
-    is-isometry-cauchy-pseudocompletion-isometry-Metric-Space :
-      is-isometry-Pseudometric-Space
-        ( cauchy-pseudocompletion-Metric-Space X)
-        ( cauchy-pseudocompletion-Metric-Space Y)
-        ( map-isometry-cauchy-approximation-Metric-Space X Y f)
-    is-isometry-cauchy-pseudocompletion-isometry-Metric-Space =
-      is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
-        ( pseudometric-Metric-Space X)
-        ( pseudometric-Metric-Space Y)
-        ( f)
-
-  isometry-cauchy-pseudocompletion-isometry-Metric-Space :
-    isometry-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Space X)
-      ( cauchy-pseudocompletion-Metric-Space Y)
-  isometry-cauchy-pseudocompletion-isometry-Metric-Space =
-    isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( pseudometric-Metric-Space Y)
-      ( f)
-```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
deleted file mode 100644
index 16d2b0789ef..00000000000
--- a/src/metric-spaces/action-on-cauchy-approximations-isometries-pseudometric-spaces.lagda.md
+++ /dev/null
@@ -1,102 +0,0 @@
-# The action on Cauchy approximations of isometries in pseudometric spaces
-
-```agda
-module metric-spaces.action-on-cauchy-approximations-isometries-pseudometric-spaces where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import elementary-number-theory.addition-positive-rational-numbers
-
-open import foundation.dependent-pair-types
-open import foundation.universe-levels
-
-open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
-open import metric-spaces.cauchy-approximations-pseudometric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
-open import metric-spaces.isometries-pseudometric-spaces
-open import metric-spaces.pseudometric-spaces
-```
-
-</details>
-
-## Idea
-
-[Isometries](metric-spaces.isometries-pseudometric-spaces.md) on
-[pseudometric spaces](metric-spaces.pseudometric-spaces.md) induce isometries on
-the
-[Cauchy pseudocompletion](metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces.md)
-of the pseudometric spaces.
-
-## Definition
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Pseudometric-Space l1 l2)
-  (Y : Pseudometric-Space l3 l4)
-  (f : isometry-Pseudometric-Space X Y)
-  where
-
-  map-isometry-cauchy-approximation-Pseudometric-Space :
-    cauchy-approximation-Pseudometric-Space X →
-    cauchy-approximation-Pseudometric-Space Y
-  map-isometry-cauchy-approximation-Pseudometric-Space =
-    map-short-map-cauchy-approximation-Pseudometric-Space
-      ( X)
-      ( Y)
-      ( short-map-isometry-Pseudometric-Space X Y f)
-```
-
-## Properties
-
-### Mapping an isometry on Cauchy approximations in a pseudometric space is an isometry in the Cauchy pseudocompletion
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Pseudometric-Space l1 l2)
-  (Y : Pseudometric-Space l3 l4)
-  (f : isometry-Pseudometric-Space X Y)
-  where
-
-  abstract
-    is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space :
-      is-isometry-Pseudometric-Space
-        ( cauchy-pseudocompletion-Pseudometric-Space X)
-        ( cauchy-pseudocompletion-Pseudometric-Space Y)
-        ( map-isometry-cauchy-approximation-Pseudometric-Space X Y f)
-    pr1
-      ( is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
-        d (x , is-approx-x) (y , is-approx-y))
-      Ndxy δ ε =
-      preserves-neighborhoods-map-isometry-Pseudometric-Space
-        ( X)
-        ( Y)
-        ( f)
-        ( δ +ℚ⁺ ε +ℚ⁺ d)
-        ( x δ)
-        ( y ε)
-        ( Ndxy δ ε)
-    pr2
-      ( is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space
-        d (x , is-approx-x) (y , is-approx-y))
-      Ndfxfy δ ε =
-      reflects-neighborhoods-map-isometry-Pseudometric-Space
-        ( X)
-        ( Y)
-        ( f)
-        ( δ +ℚ⁺ ε +ℚ⁺ d)
-        ( x δ)
-        ( y ε)
-        ( Ndfxfy δ ε)
-
-  isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space :
-    isometry-Pseudometric-Space
-      ( cauchy-pseudocompletion-Pseudometric-Space X)
-      ( cauchy-pseudocompletion-Pseudometric-Space Y)
-  isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space =
-    ( map-isometry-cauchy-approximation-Pseudometric-Space X Y f ,
-      is-isometry-cauchy-pseudocompletion-isometry-Pseudometric-Space)
-```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
deleted file mode 100644
index 0af182cc02c..00000000000
--- a/src/metric-spaces/action-on-cauchy-approximations-short-maps-metric-spaces.lagda.md
+++ /dev/null
@@ -1,59 +0,0 @@
-# The action on Cauchy approximations of short maps on metric spaces
-
-```agda
-module metric-spaces.action-on-cauchy-approximations-short-maps-metric-spaces where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import foundation.universe-levels
-
-open import metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces
-open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
-open import metric-spaces.metric-spaces
-open import metric-spaces.short-maps-metric-spaces
-open import metric-spaces.short-maps-pseudometric-spaces
-```
-
-</details>
-
-## Idea
-
-The action of [short maps](metric-spaces.short-maps-metric-spaces.md) on
-[Cauchy approximations](metric-spaces.cauchy-approximations-metric-spaces.md) in
-[metric spaces](metric-spaces.metric-spaces.md) is itself a
-[short map](metric-spaces.short-maps-pseudometric-spaces.md) on the
-[Cauchy pseudocompletions](metric-spaces.cauchy-pseudocompletions-of-metric-spaces.md)
-of the metric spaces.
-
-## Definition
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Metric-Space l1 l2)
-  (Y : Metric-Space l3 l4)
-  (f : short-map-Metric-Space X Y)
-  where
-
-  short-map-cauchy-pseudocompletion-short-map-Metric-Space :
-    short-map-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Space X)
-      ( cauchy-pseudocompletion-Metric-Space Y)
-  short-map-cauchy-pseudocompletion-short-map-Metric-Space =
-    short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( pseudometric-Metric-Space Y)
-      ( f)
-
-  map-short-map-cauchy-approximation-Metric-Space :
-    cauchy-approximation-Metric-Space X →
-    cauchy-approximation-Metric-Space Y
-  map-short-map-cauchy-approximation-Metric-Space =
-    map-short-map-cauchy-approximation-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( pseudometric-Metric-Space Y)
-      ( f)
-```
diff --git a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md b/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
deleted file mode 100644
index 281e54b3bd9..00000000000
--- a/src/metric-spaces/action-on-cauchy-approximations-short-maps-pseudometric-spaces.lagda.md
+++ /dev/null
@@ -1,92 +0,0 @@
-# The action on Cauchy approximations of short maps in pseudometric spaces
-
-```agda
-module metric-spaces.action-on-cauchy-approximations-short-maps-pseudometric-spaces where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import elementary-number-theory.addition-positive-rational-numbers
-open import elementary-number-theory.positive-rational-numbers
-
-open import foundation.dependent-pair-types
-open import foundation.universe-levels
-
-open import metric-spaces.cauchy-approximations-pseudometric-spaces
-open import metric-spaces.cauchy-pseudocompletions-of-pseudometric-spaces
-open import metric-spaces.pseudometric-spaces
-open import metric-spaces.short-maps-pseudometric-spaces
-```
-
-</details>
-
-## Idea
-
-The action of [short maps](metric-spaces.short-maps-pseudometric-spaces.md) on
-[pseudometric spaces](metric-spaces.pseudometric-spaces.md) preserves
-[Cauchy approximations](metric-spaces.cauchy-approximations-pseudometric-spaces.md).
-
-## Definition
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Pseudometric-Space l1 l2)
-  (Y : Pseudometric-Space l3 l4)
-  (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
-  (x@(map-x , is-approx-x) : cauchy-approximation-Pseudometric-Space X)
-  where
-
-  map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space :
-    ℚ⁺ → type-Pseudometric-Space Y
-  map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space
-    ε =
-    map-f (map-x ε)
-
-  abstract
-    is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space :
-      is-cauchy-approximation-Pseudometric-Space
-        ( Y)
-        ( map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space)
-    is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space
-      δ ε =
-      is-short-f (δ +ℚ⁺ ε) (map-x δ) (map-x ε) (is-approx-x δ ε)
-
-  map-short-map-cauchy-approximation-Pseudometric-Space :
-    cauchy-approximation-Pseudometric-Space Y
-  map-short-map-cauchy-approximation-Pseudometric-Space =
-    ( map-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space ,
-      is-cauchy-approximation-map-short-map-cauchy-approximation-Pseudometric-Space)
-```
-
-## Properties
-
-### The induced action of a short map on Cauchy approximations is short in the Cauchy pseudocompletion of a pseudometric space
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level}
-  (X : Pseudometric-Space l1 l2)
-  (Y : Pseudometric-Space l3 l4)
-  (f@(map-f , is-short-f) : short-map-Pseudometric-Space X Y)
-  where
-
-  abstract
-    is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space :
-      is-short-map-Pseudometric-Space
-        ( cauchy-pseudocompletion-Pseudometric-Space X)
-        ( cauchy-pseudocompletion-Pseudometric-Space Y)
-        ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f)
-    is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space
-      d (x , _) (y , _) Ndxy δ ε =
-      is-short-f (δ +ℚ⁺ ε +ℚ⁺ d) (x δ) (y ε) (Ndxy δ ε)
-
-  short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space :
-    short-map-Pseudometric-Space
-      ( cauchy-pseudocompletion-Pseudometric-Space X)
-      ( cauchy-pseudocompletion-Pseudometric-Space Y)
-  short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space =
-    ( map-short-map-cauchy-approximation-Pseudometric-Space X Y f ,
-      is-short-map-cauchy-pseudocompletion-short-map-Pseudometric-Space)
-```
diff --git a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
index 524cf3c7a21..d7627035b9f 100644
--- a/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
@@ -10,8 +10,10 @@ module metric-spaces.limits-of-cauchy-approximations-metric-spaces where
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
+open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.universe-levels
 
@@ -245,6 +247,33 @@ module _
       ( is-lim-v-y)
 ```
 
+### Cauchy approximations with limits are similar if and only if the limits are equal
+
+```agda
+module _
+  {l1 l2 : Level}
+  (M : Metric-Space l1 l2)
+  (u v : cauchy-approximation-Metric-Space M)
+  {x y : type-Metric-Space M}
+  (is-lim-u-x : is-limit-cauchy-approximation-Metric-Space M u x)
+  (is-lim-v-y : is-limit-cauchy-approximation-Metric-Space M v y)
+  where
+
+  eq-limit-iff-sim-cauchy-pseudocompletion-Metric-Space :
+    sim-cauchy-pseudocompletion-Metric-Space M u v ↔ (x ＝ y)
+  pr1 eq-limit-iff-sim-cauchy-pseudocompletion-Metric-Space u~v =
+    eq-limit-sim-cauchy-pseudocompletion-Metric-Space M
+      ( u)
+      ( v)
+      ( u~v)
+      ( x)
+      ( y)
+      ( is-lim-u-x)
+      ( is-lim-v-y)
+  pr2 eq-limit-iff-sim-cauchy-pseudocompletion-Metric-Space refl =
+    sim-is-limit-cauchy-approximation-Metric-Space M u v x is-lim-u-x is-lim-v-y
+```
+
 ### Homotopic Cauchy approximations have the same limits
 
 ```agda
@@ -278,65 +307,58 @@ module _
   (x y : type-Metric-Space X)
   (is-lim-f-x : is-limit-cauchy-approximation-Metric-Space X f x)
   (is-lim-g-y : is-limit-cauchy-approximation-Metric-Space X g y)
-  where abstract
+  where
+
+  abstract
+    same-neighborhoods-limits-cauchy-pseudocompletion-Metric-Space :
+      neighborhood-cauchy-pseudocompletion-Metric-Space X d f g ↔
+      neighborhood-Metric-Space X d x y
+    same-neighborhoods-limits-cauchy-pseudocompletion-Metric-Space =
+      logical-equivalence-reasoning
+        neighborhood-cauchy-pseudocompletion-Metric-Space X d f g
+        ↔ neighborhood-cauchy-pseudocompletion-Metric-Space X
+            ( d)
+            ( const-cauchy-approximation-Metric-Space X x)
+            ( const-cauchy-approximation-Metric-Space X y)
+          by
+            preserves-and-reflects-neighborhoods-sim-Pseudometric-Space
+              ( cauchy-pseudocompletion-Metric-Space X)
+              { x = f}
+              { x' = const-cauchy-approximation-Metric-Space X x}
+              { y = g}
+              { y' = const-cauchy-approximation-Metric-Space X y}
+              ( sim-const-is-limit-cauchy-approximation-Metric-Space X
+                ( f)
+                ( x)
+                ( is-lim-f-x))
+              ( sim-const-is-limit-cauchy-approximation-Metric-Space
+                ( X)
+                ( g)
+                ( y)
+                ( is-lim-g-y))
+              ( d)
+        ↔ neighborhood-Metric-Space X d x y
+          by
+            inv-iff
+              ( is-isometry-map-unit-cauchy-pseudocompletion-Metric-Space
+                ( X)
+                ( d)
+                ( x)
+                ( y))
 
   preserves-neighborhoods-limits-cauchy-approximation-Metric-Space :
     neighborhood-cauchy-pseudocompletion-Metric-Space X d f g →
     neighborhood-Metric-Space X d x y
-  preserves-neighborhoods-limits-cauchy-approximation-Metric-Space Ndfg =
-    reflects-neighborhoods-map-isometry-Pseudometric-Space
-      ( pseudometric-Metric-Space X)
-      ( cauchy-pseudocompletion-Metric-Space X)
-      ( isometry-unit-cauchy-pseudocompletion-Metric-Space X)
-      ( d)
-      ( x)
-      ( y)
-      ( preserves-neighborhoods-sim-Pseudometric-Space
-        ( cauchy-pseudocompletion-Metric-Space X)
-        { x = f}
-        { x' = const-cauchy-approximation-Metric-Space X x}
-        { y = g}
-        { y' = const-cauchy-approximation-Metric-Space X y}
-        ( sim-const-is-limit-cauchy-approximation-Metric-Space X f x is-lim-f-x)
-        ( sim-const-is-limit-cauchy-approximation-Metric-Space X g y is-lim-g-y)
-        ( d)
-        ( Ndfg))
+  preserves-neighborhoods-limits-cauchy-approximation-Metric-Space =
+    forward-implication
+      ( same-neighborhoods-limits-cauchy-pseudocompletion-Metric-Space)
 
   reflects-neighborhoods-limits-cauchy-approximation-Metric-Space :
     neighborhood-Metric-Space X d x y →
     neighborhood-cauchy-pseudocompletion-Metric-Space X d f g
-  reflects-neighborhoods-limits-cauchy-approximation-Metric-Space Ndxy =
-    preserves-neighborhoods-sim-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Space X)
-      { x = const-cauchy-approximation-Metric-Space X x}
-      { x' = f}
-      { y = const-cauchy-approximation-Metric-Space X y}
-      { y' = g}
-      ( symmetric-sim-Pseudometric-Space
-        ( cauchy-pseudocompletion-Metric-Space X)
-        ( f)
-        ( const-cauchy-approximation-Metric-Space X x)
-        ( sim-const-is-limit-cauchy-approximation-Metric-Space X
-          ( f)
-          ( x)
-          ( is-lim-f-x)))
-      ( symmetric-sim-Pseudometric-Space
-        ( cauchy-pseudocompletion-Metric-Space X)
-        ( g)
-        ( const-cauchy-approximation-Metric-Space X y)
-        ( sim-const-is-limit-cauchy-approximation-Metric-Space X
-          ( g)
-          ( y)
-          ( is-lim-g-y)))
-      ( d)
-      ( preserves-neighborhoods-map-isometry-Pseudometric-Space
-        ( pseudometric-Metric-Space X)
-        ( cauchy-pseudocompletion-Metric-Space X)
-        ( isometry-unit-cauchy-pseudocompletion-Metric-Space X)
-        ( d)
-        ( x)
-        ( y)
-        ( Ndxy))
+  reflects-neighborhoods-limits-cauchy-approximation-Metric-Space =
+    backward-implication
+      ( same-neighborhoods-limits-cauchy-pseudocompletion-Metric-Space)
 ```
 
 ## See also
diff --git a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
index be3f658b930..304b77c7d64 100644
--- a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
+++ b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
@@ -169,15 +169,17 @@ abstract
           ( associative-add-ℚ y (rational-ℚ⁺ ε) (rational-ℚ⁺ δ))
           ( pr2 (H δ)))
 
-pseudometric-space-ℚ : Pseudometric-Space lzero lzero
-pr1 pseudometric-space-ℚ = ℚ
-pr2 pseudometric-space-ℚ =
+pseudometric-structure-ℚ : Pseudometric-Structure lzero ℚ
+pseudometric-structure-ℚ =
   ( neighborhood-prop-ℚ ,
     is-reflexive-neighborhood-ℚ ,
     is-symmetric-neighborhood-ℚ ,
     is-triangular-neighborhood-ℚ ,
     is-saturated-neighborhood-ℚ)
 
+pseudometric-space-ℚ : Pseudometric-Space lzero lzero
+pseudometric-space-ℚ = (ℚ , pseudometric-structure-ℚ)
+
 abstract
   is-tight-pseudometric-space-ℚ :
     is-tight-Pseudometric-Space pseudometric-space-ℚ
@@ -306,6 +308,16 @@ module _
   where
 
   abstract
+    is-short-map-left-add-ℚ :
+      is-short-map-Metric-Space
+        ( metric-space-ℚ)
+        ( metric-space-ℚ)
+        ( add-ℚ x)
+    is-short-map-left-add-ℚ d y z =
+      map-product
+        ( preserves-lower-neighborhood-add-ℚ x y z d)
+        ( preserves-lower-neighborhood-add-ℚ x z y d)
+
     is-isometry-left-add-ℚ :
       is-isometry-Metric-Space
         ( metric-space-ℚ)
@@ -313,9 +325,7 @@ module _
         ( add-ℚ x)
     is-isometry-left-add-ℚ d y z =
       pair
-        ( map-product
-          ( preserves-lower-neighborhood-add-ℚ x y z d)
-          ( preserves-lower-neighborhood-add-ℚ x z y d))
+        ( is-short-map-left-add-ℚ d y z)
         ( map-product
           ( reflects-lower-neighborhood-add-ℚ x y z d)
           ( reflects-lower-neighborhood-add-ℚ x z y d))
@@ -372,6 +382,15 @@ abstract
       ↔ neighborhood-ℚ d (neg-ℚ x) (neg-ℚ y)
         by leq-dist-iff-neighborhood-ℚ _ _ _
 
+  is-short-map-neg-ℚ :
+    is-short-map-Metric-Space metric-space-ℚ metric-space-ℚ neg-ℚ
+  is-short-map-neg-ℚ =
+    is-short-map-is-isometry-Metric-Space
+      ( metric-space-ℚ)
+      ( metric-space-ℚ)
+      ( neg-ℚ)
+      ( is-isometry-neg-ℚ)
+
   is-uniformly-continuous-map-neg-ℚ :
     is-uniformly-continuous-map-Metric-Space
       ( metric-space-ℚ)
diff --git a/src/metric-spaces/similarity-of-elements-pseudometric-spaces.lagda.md b/src/metric-spaces/similarity-of-elements-pseudometric-spaces.lagda.md
index ae932073c8d..296228f80c3 100644
--- a/src/metric-spaces/similarity-of-elements-pseudometric-spaces.lagda.md
+++ b/src/metric-spaces/similarity-of-elements-pseudometric-spaces.lagda.md
@@ -269,6 +269,18 @@ module _
         backward-implication
           ( same-neighbors d _)
           ( refl-sim-Pseudometric-Space A _ d))
+
+    preserves-and-reflects-neighborhoods-sim-Pseudometric-Space :
+      {x x' y y' : type-Pseudometric-Space A} →
+      sim-Pseudometric-Space A x x' →
+      sim-Pseudometric-Space A y y' →
+      (d : ℚ⁺) →
+      neighborhood-Pseudometric-Space A d x y ↔
+      neighborhood-Pseudometric-Space A d x' y'
+    preserves-and-reflects-neighborhoods-sim-Pseudometric-Space
+      {x} {x'} {y} {y'} x~x' y~y' d =
+      ( preserves-neighborhoods-sim-Pseudometric-Space x~x' y~y' d ,
+        reflects-neighborhoods-sim-Pseudometric-Space x~x' y~y' d)
 ```
 
 ### Similar elements are elements similar w.r.t the underlying rational neighborhood relation
@@ -362,4 +374,14 @@ module _
     ( sim-Pseudometric-Space A x y)
   reflects-sim-map-isometry-Pseudometric-Space x y fx~fy d =
     reflects-neighborhoods-map-isometry-Pseudometric-Space A B f d x y (fx~fy d)
+
+  iff-sim-map-isometry-Pseudometric-Space :
+    ( x y : type-Pseudometric-Space A) →
+    ( sim-Pseudometric-Space A x y) ↔
+    ( sim-Pseudometric-Space B
+      ( map-isometry-Pseudometric-Space A B f x)
+      ( map-isometry-Pseudometric-Space A B f y))
+  iff-sim-map-isometry-Pseudometric-Space x y =
+    ( preserves-sim-map-isometry-Pseudometric-Space x y ,
+      reflects-sim-map-isometry-Pseudometric-Space x y)
 ```

From 53ada379e2adb19e90604648990dc9424076fd9a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 14 Jun 2026 11:51:13 -0700
Subject: [PATCH 24/26] Fixes

---
 ...s-maps-into-metric-abelian-groups.lagda.md | 28 ++++++------
 ...roximations-metric-abelian-groups.lagda.md | 30 +++++++------
 ...dditive-group-of-rational-numbers.lagda.md |  4 +-
 ...-groups-normed-real-vector-spaces.lagda.md |  4 +-
 .../normed-real-vector-spaces.lagda.md        | 25 +++++++++++
 .../isometry-addition-real-numbers.lagda.md   | 45 ++++++++++++-------
 .../isometry-negation-real-numbers.lagda.md   | 22 ++-------
 ...ic-additive-group-of-real-numbers.lagda.md |  4 +-
 8 files changed, 94 insertions(+), 68 deletions(-)

diff --git a/src/analysis/metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups.lagda.md
index ecce1868350..f54b856bfe8 100644
--- a/src/analysis/metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-of-uniformly-continuous-maps-into-metric-abelian-groups.lagda.md
@@ -18,11 +18,11 @@ open import group-theory.function-abelian-groups
 open import group-theory.subgroups-abelian-groups
 
 open import metric-spaces.cartesian-products-metric-spaces
-open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.maps-metric-spaces
 open import metric-spaces.metric-space-of-uniformly-continuous-maps-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.pseudometric-spaces
+open import metric-spaces.short-maps-metric-spaces
 open import metric-spaces.uniformly-continuous-maps-metric-spaces
 ```
 
@@ -154,26 +154,24 @@ module _
     add-Ab ab-uniformly-continuous-map-Metric-Ab
 
   abstract
-    is-isometry-neg-uniformly-continuous-map-Metric-Ab :
-      is-isometry-Metric-Space
+    is-short-map-neg-uniformly-continuous-map-Metric-Ab :
+      is-short-map-Metric-Space
         ( metric-space-uniformly-continuous-map-Metric-Ab)
         ( metric-space-uniformly-continuous-map-Metric-Ab)
         ( neg-uniformly-continuous-map-Metric-Ab)
-    is-isometry-neg-uniformly-continuous-map-Metric-Ab
-      d f@(map-f , _) g@(map-g , _) =
-      iff-Π-iff-family
-        ( λ x → is-isometry-neg-Metric-Ab G d (map-f x) (map-g x))
+    is-short-map-neg-uniformly-continuous-map-Metric-Ab
+      d f@(map-f , _) g@(map-g , _) Ndfg x =
+      is-short-map-neg-Metric-Ab G d (map-f x) (map-g x) (Ndfg x)
 
-    is-isometry-add-uniformly-continuous-map-Metric-Ab :
+    is-short-map-add-uniformly-continuous-map-Metric-Ab :
       (f : type-uniformly-continuous-map-Metric-Ab) →
-      is-isometry-Metric-Space
+      is-short-map-Metric-Space
         ( metric-space-uniformly-continuous-map-Metric-Ab)
         ( metric-space-uniformly-continuous-map-Metric-Ab)
         ( add-uniformly-continuous-map-Metric-Ab f)
-    is-isometry-add-uniformly-continuous-map-Metric-Ab
-      (map-f , _) d (map-g , _) (map-h , _) =
-      iff-Π-iff-family
-        ( λ x → is-isometry-add-Metric-Ab G (map-f x) d (map-g x) (map-h x))
+    is-short-map-add-uniformly-continuous-map-Metric-Ab
+      (map-f , _) d (map-g , _) (map-h , _) Ndgh x =
+      is-short-map-add-Metric-Ab G (map-f x) d (map-g x) (map-h x) (Ndgh x)
 
   metric-ab-uniformly-continuous-map-Metric-Ab :
     Metric-Ab (l1 ⊔ l2 ⊔ l3 ⊔ l4) (l1 ⊔ l4)
@@ -182,6 +180,6 @@ module _
       pseudometric-structure-uniformly-continuous-map-Metric-Ab ,
       is-extensional-pseudometric-Metric-Space
         ( metric-space-uniformly-continuous-map-Metric-Ab) ,
-      is-isometry-neg-uniformly-continuous-map-Metric-Ab ,
-      is-isometry-add-uniformly-continuous-map-Metric-Ab)
+      is-short-map-neg-uniformly-continuous-map-Metric-Ab ,
+      is-short-map-add-uniformly-continuous-map-Metric-Ab)
 ```
diff --git a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
index 6d7634281e3..51c27be6649 100644
--- a/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
+++ b/src/analysis/negation-cauchy-approximations-metric-abelian-groups.lagda.md
@@ -18,7 +18,7 @@ open import foundation.identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 
-open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.functoriality-isometries-cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.isometries-pseudometric-spaces
 ```
 
@@ -43,7 +43,7 @@ module _
   neg-cauchy-approximation-Metric-Ab :
     cauchy-approximation-Metric-Ab G → cauchy-approximation-Metric-Ab G
   neg-cauchy-approximation-Metric-Ab =
-    map-isometry-cauchy-approximation-Metric-Space
+    map-isometry-cauchy-pseudocompletion-Metric-Space
       ( metric-space-Metric-Ab G)
       ( metric-space-Metric-Ab G)
       ( isometry-neg-Metric-Ab G)
@@ -59,6 +59,16 @@ module _
   (G : Metric-Ab l1 l2)
   where
 
+  isometry-neg-cauchy-pseudocompletion-Metric-Ab :
+    isometry-Pseudometric-Space
+      ( cauchy-pseudocompletion-Metric-Ab G)
+      ( cauchy-pseudocompletion-Metric-Ab G)
+  isometry-neg-cauchy-pseudocompletion-Metric-Ab =
+    isometry-cauchy-pseudocompletion-Metric-Space
+      ( metric-space-Metric-Ab G)
+      ( metric-space-Metric-Ab G)
+      ( isometry-neg-Metric-Ab G)
+
   abstract
     is-isometry-neg-cauchy-pseudocompletion-Metric-Ab :
       is-isometry-Pseudometric-Space
@@ -66,18 +76,10 @@ module _
         ( cauchy-pseudocompletion-Metric-Ab G)
         ( neg-cauchy-approximation-Metric-Ab G)
     is-isometry-neg-cauchy-pseudocompletion-Metric-Ab =
-      is-isometry-cauchy-pseudocompletion-isometry-Metric-Space
-        ( metric-space-Metric-Ab G)
-        ( metric-space-Metric-Ab G)
-        ( isometry-neg-Metric-Ab G)
-
-  isometry-neg-cauchy-pseudocompletion-Metric-Ab :
-    isometry-Pseudometric-Space
-      ( cauchy-pseudocompletion-Metric-Ab G)
-      ( cauchy-pseudocompletion-Metric-Ab G)
-  isometry-neg-cauchy-pseudocompletion-Metric-Ab =
-    ( neg-cauchy-approximation-Metric-Ab G ,
-      is-isometry-neg-cauchy-pseudocompletion-Metric-Ab)
+      is-isometry-map-isometry-Pseudometric-Space
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( cauchy-pseudocompletion-Metric-Ab G)
+        ( isometry-neg-cauchy-pseudocompletion-Metric-Ab)
 ```
 
 ### Inverse laws of addition of Cauchy approximations
diff --git a/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
index 38d71c14497..aa78014a8b5 100644
--- a/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
@@ -34,6 +34,6 @@ metric-ab-add-ℚ =
   ( abelian-group-add-ℚ ,
     pseudometric-structure-Metric-Space metric-space-ℚ ,
     is-extensional-pseudometric-space-ℚ ,
-    is-isometry-neg-ℚ ,
-    is-isometry-left-add-ℚ)
+    is-short-map-neg-ℚ ,
+    is-short-map-left-add-ℚ)
 ```
diff --git a/src/functional-analysis/metric-abelian-groups-normed-real-vector-spaces.lagda.md b/src/functional-analysis/metric-abelian-groups-normed-real-vector-spaces.lagda.md
index afd9de40317..9e452b93110 100644
--- a/src/functional-analysis/metric-abelian-groups-normed-real-vector-spaces.lagda.md
+++ b/src/functional-analysis/metric-abelian-groups-normed-real-vector-spaces.lagda.md
@@ -69,8 +69,8 @@ module _
       pseudometric-structure-metric-ab-Normed-ℝ-Vector-Space ,
       is-extensional-pseudometric-Metric-Space
         ( metric-space-Normed-ℝ-Vector-Space V) ,
-      is-isometry-neg-Normed-ℝ-Vector-Space V ,
-      is-isometry-left-add-Normed-ℝ-Vector-Space V)
+      is-short-map-neg-Normed-ℝ-Vector-Space V ,
+      is-short-map-left-add-Normed-ℝ-Vector-Space V)
 ```
 
 ## Properties
diff --git a/src/linear-algebra/normed-real-vector-spaces.lagda.md b/src/linear-algebra/normed-real-vector-spaces.lagda.md
index 53a58db9b70..29c56236d5b 100644
--- a/src/linear-algebra/normed-real-vector-spaces.lagda.md
+++ b/src/linear-algebra/normed-real-vector-spaces.lagda.md
@@ -31,6 +31,7 @@ open import metric-spaces.located-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.metrics
 open import metric-spaces.metrics-of-metric-spaces
+open import metric-spaces.short-maps-metric-spaces
 
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
@@ -363,6 +364,18 @@ module _
                         ( _))
                 ＝ dist-Normed-ℝ-Vector-Space V x y
                   by symmetric-dist-Normed-ℝ-Vector-Space V y x)))
+
+    is-short-map-neg-Normed-ℝ-Vector-Space :
+      is-short-map-Metric-Space
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( neg-Normed-ℝ-Vector-Space V)
+    is-short-map-neg-Normed-ℝ-Vector-Space =
+      is-short-map-is-isometry-Metric-Space
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( neg-Normed-ℝ-Vector-Space V)
+        ( is-isometry-neg-Normed-ℝ-Vector-Space)
 ```
 
 ### Left addition is an isometry in the metric space of a normed vector space
@@ -403,6 +416,18 @@ module _
                   ( u)
                   ( v)
                   ( w)))))
+
+    is-short-map-left-add-Normed-ℝ-Vector-Space :
+      is-short-map-Metric-Space
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( add-Normed-ℝ-Vector-Space V u)
+    is-short-map-left-add-Normed-ℝ-Vector-Space =
+      is-short-map-is-isometry-Metric-Space
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( metric-space-Normed-ℝ-Vector-Space V)
+        ( add-Normed-ℝ-Vector-Space V u)
+        ( is-isometry-left-add-Normed-ℝ-Vector-Space)
 ```
 
 ### The norm of the zero vector is zero
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index 55b9f3b3e51..be0a83e6e90 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -15,6 +15,7 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cartesian-products-metric-spaces
+open import metric-spaces.expansive-maps-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-space-of-isometries-metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-maps-metric-spaces
@@ -51,27 +52,41 @@ module _
   where
 
   abstract
+    is-short-map-left-add-ℝ :
+      is-short-map-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( add-ℝ x)
+    is-short-map-left-add-ℝ d y z Nyz =
+      neighborhood-real-bound-each-leq-ℝ
+        ( d)
+        ( add-ℝ x y)
+        ( add-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-add-ℝ d x y z
+          ( left-leq-real-bound-neighborhood-ℝ d y z Nyz))
+        ( preserves-lower-neighborhood-leq-left-add-ℝ d x z y
+          ( right-leq-real-bound-neighborhood-ℝ d y z Nyz))
+
+    is-expansive-map-left-add-ℝ :
+      is-expansive-map-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( add-ℝ x)
+    is-expansive-map-left-add-ℝ d y z Nxyz =
+      neighborhood-real-bound-each-leq-ℝ d y z
+        ( reflects-lower-neighborhood-leq-left-add-ℝ d x y z
+          ( left-leq-real-bound-neighborhood-ℝ d (x +ℝ y) (x +ℝ z) Nxyz))
+        ( reflects-lower-neighborhood-leq-left-add-ℝ d x z y
+          ( right-leq-real-bound-neighborhood-ℝ d (x +ℝ y) (x +ℝ z) Nxyz))
+
     is-isometry-left-add-ℝ :
       is-isometry-Metric-Space
         ( metric-space-ℝ l2)
         ( metric-space-ℝ (l1 ⊔ l2))
         ( add-ℝ x)
     is-isometry-left-add-ℝ d y z =
-      ( λ Nyz →
-        neighborhood-real-bound-each-leq-ℝ
-          ( d)
-          ( add-ℝ x y)
-          ( add-ℝ x z)
-          ( preserves-lower-neighborhood-leq-left-add-ℝ d x y z
-            ( left-leq-real-bound-neighborhood-ℝ d y z Nyz))
-          ( preserves-lower-neighborhood-leq-left-add-ℝ d x z y
-            ( right-leq-real-bound-neighborhood-ℝ d y z Nyz))) ,
-      ( λ Nxyz →
-        neighborhood-real-bound-each-leq-ℝ d y z
-          ( reflects-lower-neighborhood-leq-left-add-ℝ d x y z
-            ( left-leq-real-bound-neighborhood-ℝ d (x +ℝ y) (x +ℝ z) Nxyz))
-          ( reflects-lower-neighborhood-leq-left-add-ℝ d x z y
-            ( right-leq-real-bound-neighborhood-ℝ d (x +ℝ y) (x +ℝ z) Nxyz)))
+      ( is-short-map-left-add-ℝ d y z ,
+        is-expansive-map-left-add-ℝ d y z)
 
     is-isometry-right-add-ℝ :
       is-isometry-Metric-Space
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index c12aba0caf9..022ff28dd7c 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -74,10 +74,10 @@ module _
   where
 
   abstract
-    neg-neighborhood-ℝ : (d : ℚ⁺) (x y : ℝ l1) →
+    is-short-map-neg-ℝ : (d : ℚ⁺) (x y : ℝ l1) →
       neighborhood-ℝ l1 d x y →
       neighborhood-ℝ l1 d (neg-ℝ x) (neg-ℝ y)
-    neg-neighborhood-ℝ d x y H =
+    is-short-map-neg-ℝ d x y H =
       neighborhood-real-bound-each-leq-ℝ
         ( d)
         ( neg-ℝ x)
@@ -108,12 +108,12 @@ module _
         ( metric-space-ℝ l1)
         ( neg-ℝ)
     is-isometry-neg-ℝ d x y =
-      ( neg-neighborhood-ℝ d x y) ,
+      ( is-short-map-neg-ℝ d x y) ,
       ( ( binary-tr
           ( neighborhood-ℝ l1 d)
           ( neg-neg-ℝ x)
           ( neg-neg-ℝ y)) ∘
-        ( neg-neighborhood-ℝ
+        ( is-short-map-neg-ℝ
           ( d)
           ( neg-ℝ x)
           ( neg-ℝ y)))
@@ -128,20 +128,6 @@ module _
 ### Negation on the real numbers is short
 
 ```agda
-abstract
-  is-short-map-neg-ℝ :
-    {l : Level} →
-    is-short-map-Metric-Space
-      ( metric-space-ℝ l)
-      ( metric-space-ℝ l)
-      ( neg-ℝ)
-  is-short-map-neg-ℝ =
-    is-short-map-is-isometry-Metric-Space
-      ( metric-space-ℝ _)
-      ( metric-space-ℝ _)
-      ( neg-ℝ)
-      ( is-isometry-neg-ℝ)
-
 short-map-neg-ℝ :
   {l : Level} →
   short-map-Metric-Space
diff --git a/src/real-numbers/metric-additive-group-of-real-numbers.lagda.md b/src/real-numbers/metric-additive-group-of-real-numbers.lagda.md
index db74d26fd06..1040e36dc20 100644
--- a/src/real-numbers/metric-additive-group-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-additive-group-of-real-numbers.lagda.md
@@ -41,8 +41,8 @@ metric-ab-add-ℝ l =
   ( ab-add-ℝ l ,
     structure-Pseudometric-Space (pseudometric-space-ℝ l) ,
     is-extensional-pseudometric-space-ℝ ,
-    is-isometry-neg-ℝ ,
-    is-isometry-left-add-ℝ)
+    is-short-map-neg-ℝ ,
+    is-short-map-left-add-ℝ)
 
 complete-metric-ab-add-ℝ : (l : Level) → Complete-Metric-Ab (lsuc l) l
 complete-metric-ab-add-ℝ l =

From 0eebabac4e9fbc20ff446f512a5037a53c652a02 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 14 Jun 2026 11:52:59 -0700
Subject: [PATCH 25/26] Fix to use functoriality

---
 ...ts-cauchy-pseudocompletions-metric-abelian-groups.lagda.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 30c1bf7a15a..9dc04d2eae3 100644
--- a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -29,7 +29,7 @@ open import foundation.universe-levels
 open import group-theory.abelian-groups
 open import group-theory.homomorphisms-abelian-groups
 
-open import metric-spaces.action-on-cauchy-approximations-isometries-metric-spaces
+open import metric-spaces.functoriality-isometries-cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.isometries-pseudometric-spaces
@@ -106,7 +106,7 @@ module _
               ( metric-quotient-cauchy-pseudocompletion-Metric-Ab G))
             ( isometry-unit-metric-quotient-Pseudometric-Space
               ( cauchy-pseudocompletion-Metric-Ab G))
-            ( isometry-cauchy-pseudocompletion-isometry-Metric-Space
+            ( isometry-cauchy-pseudocompletion-Metric-Space
               ( metric-space-Metric-Ab G)
               ( metric-space-Metric-Ab G)
               ( isometry-neg-Metric-Ab G)))

From 6ddf61a8d9e231a750cc469e402853463862067c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 14 Jun 2026 11:55:54 -0700
Subject: [PATCH 26/26] make pre-commit

---
 ...ents-cauchy-pseudocompletions-metric-abelian-groups.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
index 9dc04d2eae3..5a56b5722ef 100644
--- a/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
+++ b/src/analysis/metric-abelian-groups-metric-quotients-cauchy-pseudocompletions-metric-abelian-groups.lagda.md
@@ -29,8 +29,8 @@ open import foundation.universe-levels
 open import group-theory.abelian-groups
 open import group-theory.homomorphisms-abelian-groups
 
-open import metric-spaces.functoriality-isometries-cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.cauchy-pseudocompletions-of-metric-spaces
+open import metric-spaces.functoriality-isometries-cauchy-pseudocompletions-of-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.isometries-pseudometric-spaces
 open import metric-spaces.metric-spaces