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  • src/Categories/Bicategory/Construction/Bimodules/Tensorproduct

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src/Categories/Bicategory/Construction/Bimodules/Tensorproduct/Associator.agda

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@@ -359,7 +359,7 @@ module Linear-Left where
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open Categories.Morphism.Reasoning (hom (C M₁) (C M₄)) using (pushˡ; glue)
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actionˡSq-◽∘⦃◽⊗◽⦄ : F B₃ ▷ arr (CoeqBimods B₂ B₁) ∘ᵥ actionˡ-◽∘⦃◽∘◽⦄
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≈ actionˡ-∘ B₃ (B₂ ⊗₀ B₁) ∘ᵥ (F B₃ ▷ arr (CoeqBimods B₂ B₁)) ◁ T M₁
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≈ actionˡ-∘ B₃ (B₂ ⊗₀ B₁) ∘ᵥ (F B₃ ▷ arr (CoeqBimods B₂ B₁)) ◁ T M₁
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actionˡSq-◽∘⦃◽⊗◽⦄ = glue′ (▷-resp-sq (actionˡSq-⊗ B₂ B₁)) (⟺ α⇒-▷-◁)
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where
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open hom.HomReasoning
@@ -381,7 +381,7 @@ module Linear-Left where
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open TensorproductOfBimodules.Left-Action using (actionˡSq-⊗)
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actionˡSq-⦃◽⊗◽⦄∘◽ : arr (CoeqBimods B₃ B₂) ◁ F B₁ ∘ᵥ actionˡ-⦃◽∘◽⦄∘◽
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≈ actionˡ-∘ (B₃ ⊗₀ B₂) B₁ ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁ ◁ T M₁
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≈ actionˡ-∘ (B₃ ⊗₀ B₂) B₁ ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁ ◁ T M₁
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actionˡSq-⦃◽⊗◽⦄∘◽ = glue′ (⟺ ◁-▷-exchg) (⟺ α⇒-◁-∘₁)
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where
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open hom.HomReasoning
@@ -540,7 +540,7 @@ module Linear-Right where
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where
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open Categories.Morphism.Reasoning (hom (C M₁) (C M₄)) using (glue)
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open TensorproductOfBimodules.Right-Action using (actionʳSq-⊗)
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actionʳSq-⦃◽⊗◽⦄∘◽ : arr (CoeqBimods B₃ B₂) ◁ F B₁ ∘ᵥ actionʳ-⦃◽∘◽⦄∘◽
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≈ actionʳ-∘ (B₃ ⊗₀ B₂) B₁ ∘ᵥ T M₄ ▷ (arr (CoeqBimods B₃ B₂) ◁ F B₁)
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actionʳSq-⦃◽⊗◽⦄∘◽ = glue′ (◁-resp-sq (actionʳSq-⊗ B₃ B₂)) (⟺ α⇐-▷-◁)
@@ -584,28 +584,24 @@ module Linear-Right where
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(actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
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∘ᵥ T M₄ ▷ α⇒-⊗)
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∘ᵥ T M₄ ▷
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(arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
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∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁)
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∘ᵥ T M₄ ▷ (arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
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∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁)
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≈⟨ pullʳ ∘ᵥ-distr-▷ ⟩
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actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
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∘ᵥ T M₄ ▷
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(α⇒-⊗
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∘ᵥ T M₄ ▷ (α⇒-⊗
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∘ᵥ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
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∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁)
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≈⟨ refl⟩∘⟨ ▷-resp-≈ (⟺ hexagon) ⟩
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actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
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∘ᵥ T M₄ ▷
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(arr (CoeqBimods B₃ (B₂ ⊗₀ B₁))
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∘ᵥ T M₄ ▷ (arr (CoeqBimods B₃ (B₂ ⊗₀ B₁))
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∘ᵥ F B₃ ▷ arr (CoeqBimods B₂ B₁)
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∘ᵥ α⇒) ≈⟨ refl⟩∘⟨ ⟺ ∘ᵥ-distr-▷ ⟩
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actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
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∘ᵥ T M₄ ▷ arr (CoeqBimods B₃ (B₂ ⊗₀ B₁))
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∘ᵥ T M₄ ▷
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(F B₃ ▷ arr (CoeqBimods B₂ B₁)
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∘ᵥ T M₄ ▷ (F B₃ ▷ arr (CoeqBimods B₂ B₁)
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∘ᵥ α⇒) ≈⟨ ⟺ (pull-last ∘ᵥ-distr-▷) ⟩
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(actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
@@ -649,17 +645,17 @@ module Linear-Right where
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where
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open hom.HomReasoning
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open Categories.Morphism.Reasoning (hom (C M₁) (C M₄))
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using (pullʳ; pushʳ; pull-last; glue′; glue; pull-center; extendˡ; pushˡ; assoc²εα)
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using (pullʳ; pushʳ; pull-last; glue′; glue; pull-center; extendˡ; pushˡ)
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open TensorproductOfBimodules.Right-Action using (actionʳSq-⊗)
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abstract
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linearʳ-∘arr : (actionʳ-⊗ B₃ (B₂ ⊗₀ B₁) ∘ᵥ T M₄ ▷ α⇒-⊗) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
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≈ (α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
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≈ (α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
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linearʳ-∘arr = Coequalizer⇒Epi
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(T M₄ ▷-coeq ((CoeqBimods B₃ B₂) coeq-◁ F B₁))
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((actionʳ-⊗ B₃ (B₂ ⊗₀ B₁) ∘ᵥ T M₄ ▷ α⇒-⊗) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
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((α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
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linearʳ-∘arr²
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(T M₄ ▷-coeq ((CoeqBimods B₃ B₂) coeq-◁ F B₁))
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((actionʳ-⊗ B₃ (B₂ ⊗₀ B₁) ∘ᵥ T M₄ ▷ α⇒-⊗) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
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((α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
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linearʳ-∘arr²
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where
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open Categories.Diagram.Coequalizer (hom (C M₁) (C M₄)) using (Coequalizer⇒Epi)
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