Library UniMath.Bicategories.Modifications.Examples.Associativity
Associativity laws are inverses
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Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Base.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map1Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map2Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Identitor.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Compositor.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.
Require Import UniMath.Bicategories.Transformations.PseudoTransformation.
Require Import UniMath.Bicategories.Transformations.Examples.Associativity.
Require Import UniMath.Bicategories.Modifications.Modification.
Local Open Scope cat.
Section Associativity.
Context {B₁ B₂ B₃ B₄: bicat}.
Variable (F₁ : psfunctor B₁ B₂)
(F₂ : psfunctor B₂ B₃)
(F₃ : psfunctor B₃ B₄).
Definition lassociator_rassociator_pstrans_data
: invertible_modification_data
(comp_pstrans
(lassociator_pstrans F₁ F₂ F₃)
(rassociator_pstrans F₁ F₂ F₃))
(id_pstrans _).
Proof.
intros X.
use make_invertible_2cell.
- exact (lunitor _).
- is_iso.
Defined.
Definition lassociator_rassociator_pstrans_modification
: is_modification lassociator_rassociator_pstrans_data.
Proof.
intros X Y f ; cbn.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
rewrite lunitor_lwhisker.
rewrite <- rwhisker_vcomp.
rewrite !vassocl.
etrans.
{
do 2 apply maponpaths.
rewrite !vassocr.
rewrite lunitor_triangle.
apply maponpaths_2.
rewrite !vassocl.
rewrite rwhisker_hcomp.
rewrite <- triangle_r_inv.
rewrite <- lwhisker_hcomp.
apply idpath.
}
rewrite !vassocl.
rewrite lwhisker_vcomp.
rewrite linvunitor_lunitor, lwhisker_id2, id2_right.
rewrite vcomp_lunitor.
rewrite runitor_lunitor_identity.
apply idpath.
Qed.
Definition lassociator_rassociator_pstrans
: invertible_modification
(comp_pstrans
(lassociator_pstrans F₁ F₂ F₃)
(rassociator_pstrans F₁ F₂ F₃))
(id_pstrans _).
Proof.
use make_invertible_modification.
- exact lassociator_rassociator_pstrans_data.
- exact lassociator_rassociator_pstrans_modification.
Defined.
Definition rassociator_lassociator_pstrans_data
: invertible_modification_data
(comp_pstrans
(rassociator_pstrans F₁ F₂ F₃)
(lassociator_pstrans F₁ F₂ F₃))
(id_pstrans _).
Proof.
intros X.
use make_invertible_2cell.
- exact (lunitor _).
- is_iso.
Defined.
Definition rassociator_lassociator_pstrans_is_modification
: is_modification rassociator_lassociator_pstrans_data.
Proof.
intros X Y f ; cbn.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
rewrite lunitor_lwhisker.
rewrite <- rwhisker_vcomp.
rewrite !vassocl.
etrans.
{
do 2 apply maponpaths.
rewrite !vassocr.
rewrite lunitor_triangle.
apply maponpaths_2.
rewrite !vassocl.
rewrite rwhisker_hcomp.
rewrite <- triangle_r_inv.
rewrite <- lwhisker_hcomp.
apply idpath.
}
rewrite !vassocl.
rewrite lwhisker_vcomp.
rewrite linvunitor_lunitor, lwhisker_id2, id2_right.
rewrite vcomp_lunitor.
rewrite runitor_lunitor_identity.
apply idpath.
Qed.
Definition rassociator_lassociator_pstrans
: invertible_modification
(comp_pstrans
(rassociator_pstrans F₁ F₂ F₃)
(lassociator_pstrans F₁ F₂ F₃))
(id_pstrans _).
Proof.
use make_invertible_modification.
- exact rassociator_lassociator_pstrans_data.
- exact rassociator_lassociator_pstrans_is_modification.
Defined.
End Associativity.