@@ -359,7 +359,7 @@ module Linear-Left where
359359 open Categories.Morphism.Reasoning (hom (C M₁) (C M₄)) using (pushˡ; glue)
360360
361361 actionˡSq-◽∘⦃◽⊗◽⦄ : F B₃ ▷ arr (CoeqBimods B₂ B₁) ∘ᵥ actionˡ-◽∘⦃◽∘◽⦄
362- ≈ actionˡ-∘ B₃ (B₂ ⊗₀ B₁) ∘ᵥ (F B₃ ▷ arr (CoeqBimods B₂ B₁)) ◁ T M₁
362+ ≈ actionˡ-∘ B₃ (B₂ ⊗₀ B₁) ∘ᵥ (F B₃ ▷ arr (CoeqBimods B₂ B₁)) ◁ T M₁
363363 actionˡSq-◽∘⦃◽⊗◽⦄ = glue′ (▷-resp-sq (actionˡSq-⊗ B₂ B₁)) (⟺ α⇒-▷-◁)
364364 where
365365 open hom.HomReasoning
@@ -381,7 +381,7 @@ module Linear-Left where
381381 open TensorproductOfBimodules.Left-Action using (actionˡSq-⊗)
382382
383383 actionˡSq-⦃◽⊗◽⦄∘◽ : arr (CoeqBimods B₃ B₂) ◁ F B₁ ∘ᵥ actionˡ-⦃◽∘◽⦄∘◽
384- ≈ actionˡ-∘ (B₃ ⊗₀ B₂) B₁ ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁ ◁ T M₁
384+ ≈ actionˡ-∘ (B₃ ⊗₀ B₂) B₁ ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁ ◁ T M₁
385385 actionˡSq-⦃◽⊗◽⦄∘◽ = glue′ (⟺ ◁-▷-exchg) (⟺ α⇒-◁-∘₁)
386386 where
387387 open hom.HomReasoning
@@ -540,7 +540,7 @@ module Linear-Right where
540540 where
541541 open Categories.Morphism.Reasoning (hom (C M₁) (C M₄)) using (glue)
542542 open TensorproductOfBimodules.Right-Action using (actionʳSq-⊗)
543-
543+
544544 actionʳSq-⦃◽⊗◽⦄∘◽ : arr (CoeqBimods B₃ B₂) ◁ F B₁ ∘ᵥ actionʳ-⦃◽∘◽⦄∘◽
545545 ≈ actionʳ-∘ (B₃ ⊗₀ B₂) B₁ ∘ᵥ T M₄ ▷ (arr (CoeqBimods B₃ B₂) ◁ F B₁)
546546 actionʳSq-⦃◽⊗◽⦄∘◽ = glue′ (◁-resp-sq (actionʳSq-⊗ B₃ B₂)) (⟺ α⇐-▷-◁)
@@ -584,28 +584,24 @@ module Linear-Right where
584584
585585 (actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
586586 ∘ᵥ T M₄ ▷ α⇒-⊗)
587- ∘ᵥ T M₄ ▷
588- (arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
589- ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁)
587+ ∘ᵥ T M₄ ▷ (arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
588+ ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁)
590589 ≈⟨ pullʳ ∘ᵥ-distr-▷ ⟩
591590
592591 actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
593- ∘ᵥ T M₄ ▷
594- (α⇒-⊗
592+ ∘ᵥ T M₄ ▷ (α⇒-⊗
595593 ∘ᵥ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
596594 ∘ᵥ arr (CoeqBimods B₃ B₂) ◁ F B₁)
597595 ≈⟨ refl⟩∘⟨ ▷-resp-≈ (⟺ hexagon) ⟩
598596
599597 actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
600- ∘ᵥ T M₄ ▷
601- (arr (CoeqBimods B₃ (B₂ ⊗₀ B₁))
598+ ∘ᵥ T M₄ ▷ (arr (CoeqBimods B₃ (B₂ ⊗₀ B₁))
602599 ∘ᵥ F B₃ ▷ arr (CoeqBimods B₂ B₁)
603600 ∘ᵥ α⇒) ≈⟨ refl⟩∘⟨ ⟺ ∘ᵥ-distr-▷ ⟩
604601
605602 actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
606603 ∘ᵥ T M₄ ▷ arr (CoeqBimods B₃ (B₂ ⊗₀ B₁))
607- ∘ᵥ T M₄ ▷
608- (F B₃ ▷ arr (CoeqBimods B₂ B₁)
604+ ∘ᵥ T M₄ ▷ (F B₃ ▷ arr (CoeqBimods B₂ B₁)
609605 ∘ᵥ α⇒) ≈⟨ ⟺ (pull-last ∘ᵥ-distr-▷) ⟩
610606
611607 (actionʳ-⊗ B₃ (B₂ ⊗₀ B₁)
@@ -649,17 +645,17 @@ module Linear-Right where
649645 where
650646 open hom.HomReasoning
651647 open Categories.Morphism.Reasoning (hom (C M₁) (C M₄))
652- using (pullʳ; pushʳ; pull-last; glue′; glue; pull-center; extendˡ; pushˡ; assoc²εα )
648+ using (pullʳ; pushʳ; pull-last; glue′; glue; pull-center; extendˡ; pushˡ)
653649 open TensorproductOfBimodules.Right-Action using (actionʳSq-⊗)
654650
655651 abstract
656652 linearʳ-∘arr : (actionʳ-⊗ B₃ (B₂ ⊗₀ B₁) ∘ᵥ T M₄ ▷ α⇒-⊗) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
657- ≈ (α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
653+ ≈ (α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁)
658654 linearʳ-∘arr = Coequalizer⇒Epi
659- (T M₄ ▷-coeq ((CoeqBimods B₃ B₂) coeq-◁ F B₁))
660- ((actionʳ-⊗ B₃ (B₂ ⊗₀ B₁) ∘ᵥ T M₄ ▷ α⇒-⊗) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
661- ((α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
662- linearʳ-∘arr²
655+ (T M₄ ▷-coeq ((CoeqBimods B₃ B₂) coeq-◁ F B₁))
656+ ((actionʳ-⊗ B₃ (B₂ ⊗₀ B₁) ∘ᵥ T M₄ ▷ α⇒-⊗) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
657+ ((α⇒-⊗ ∘ᵥ actionʳ-⊗ (B₃ ⊗₀ B₂) B₁) ∘ᵥ T M₄ ▷ arr (CoeqBimods (B₃ ⊗₀ B₂) B₁))
658+ linearʳ-∘arr²
663659 where
664660 open Categories.Diagram.Coequalizer (hom (C M₁) (C M₄)) using (Coequalizer⇒Epi)
665661
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