Library UniMath.CategoryTheory.Bicategories.Bicategories.BicategoryLaws
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Bicat. Import Notations.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Invertible_2cells.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Unitors.
Local Open Scope bicategory_scope.
Local Open Scope cat.
Section laws.
Context {C : bicat}.
Definition triangle_r
{X Y Z : C}
(g : C⟦Y,Z⟧)
(f : C⟦X,Y⟧)
: lunitor g ⋆⋆ id₂ f = (id₂ g ⋆⋆ runitor f) o lassociator f (id₁ Y) g.
Show proof.
Definition interchange
{X Y Z : C}
{f₁ g₁ h₁ : C⟦Y,Z⟧}
{f₂ g₂ h₂ : C⟦X,Y⟧}
(η₁ : f₁ ==> g₁) (η₂ : f₂ ==> g₂)
(ε₁ : g₁ ==> h₁) (ε₂ : g₂ ==> h₂)
: (ε₁ o η₁) ⋆⋆ (ε₂ o η₂) = (ε₁ ⋆⋆ ε₂) o (η₁ ⋆⋆ η₂).
Show proof.
Definition rinvunitor_natural
{X Y : C}
{f g : C⟦X, Y⟧}
(η : f ==> g)
: rinvunitor g o η = (id₂ (id₁ Y) ⋆⋆ η) o rinvunitor f.
Show proof.
Definition linvunitor_natural
{X Y : C}
{f g : C⟦X, Y⟧}
(η : f ==> g)
: linvunitor g o η = (η ⋆⋆ id₂ (id₁ X)) o linvunitor f.
Show proof.
Definition lwhisker_hcomp
{X Y Z : C}
{f g : C⟦Y,Z⟧}
(h : C⟦X, Y⟧)
(α : f ==> g)
: h ◃ α = id₂ h ⋆ α.
Show proof.
Definition rwhisker_hcomp
{X Y Z : C}
{f g : C⟦X,Y⟧}
(h : C⟦Y,Z⟧)
(α : f ==> g)
: α ▹ h = α ⋆ id₂ h.
Show proof.
Definition inverse_pentagon
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator f g (k ∘ h) o rassociator (g ∘ f) h k
=
(id₂ f ⋆ rassociator g h k) o (rassociator f (h ∘ g) k)
o (rassociator f g h ⋆ id₂ k).
Show proof.
Definition inverse_pentagon_2
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator (g ∘ f) h k o (lassociator f g h ⋆ id2 k)
=
lassociator f g (k ∘ h) o (f ◃ rassociator g h k)
o rassociator f (h ∘ g) k.
Show proof.
Definition inverse_pentagon_3
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator f g (k ∘ h) o rassociator (g ∘ f) h k o (id₂ k ⋆⋆ lassociator f g h)
=
rassociator g h k ⋆⋆ id₂ f o rassociator f (h ∘ g) k.
Show proof.
Definition inverse_pentagon_4
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: (lassociator g h k ⋆⋆ id₂ f) o rassociator f g (k ∘ h)
=
rassociator f (h ∘ g) k o id₂ k ⋆⋆ rassociator f g h o lassociator (g ∘ f) h k.
Show proof.
Definition inverse_pentagon_5
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: lassociator f g (k ∘ h) o (rassociator g h k ⋆⋆ id₂ f)
=
rassociator (g ∘ f) h k o (id₂ k ⋆⋆ lassociator f g h) o lassociator f (h ∘ g) k.
Show proof.
Definition inverse_pentagon_6
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator f (h ∘ g) k o id₂ k ⋆⋆ rassociator f g h
=
lassociator g h k ⋆⋆ id₂ f o rassociator f g (k ∘ h) o rassociator (g ∘ f) h k.
Show proof.
Definition pentagon_2
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: lassociator f (h ∘ g) k o lassociator g h k ⋆⋆ id₂ f
=
id₂ k ⋆⋆ rassociator f g h o lassociator (g ∘ f) h k o lassociator f g (k ∘ h).
Show proof.
Definition triangle_r_inv
{X Y Z : C}
(g : C ⟦ Y, Z ⟧) (f : C ⟦ X, Y ⟧)
: linvunitor g ⋆⋆ id₂ f
=
rassociator _ _ _ o id₂ g ⋆⋆ rinvunitor f.
Show proof.
Definition triangle_l
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: lunitor g ⋆⋆ id₂ f o rassociator _ _ _ = id₂ g ⋆⋆ runitor f.
Show proof.
Definition bc_whisker_r_compose
{X Y Z : C}
(f : C⟦X,Y⟧)
{g₁ g₂ g₃ : C⟦Y,Z⟧}
(p₁ : g₁ ==> g₂) (p₂ : g₂ ==> g₃)
: (p₂ o p₁) ▻ f = (p₂ ▻ f) o (p₁ ▻ f).
Show proof.
Definition bc_whisker_l_compose
{X Y Z : C}
{f₁ f₂ f₃ : C⟦X,Y⟧}
(g : C⟦Y,Z⟧)
(p₁ : f₁ ==> f₂) (p₂ : f₂ ==> f₃)
: g ◅ (p₂ o p₁) = (g ◅ p₂) o (g ◅ p₁).
Show proof.
Definition whisker_l_hcomp
{W X Y Z : C}
{f : C⟦X,Y⟧} {g : C⟦Y,Z⟧}
(k₁ k₂ : C⟦W,X⟧)
(α : k₁ ==> k₂)
: lassociator _ _ _ o (g ∘ f ◅ α) = g ◅ (f ◅ α) o lassociator _ _ _.
Show proof.
Definition whisker_r_hcomp
{W X Y Z : C}
{f : C⟦X,Y⟧} {g : C⟦Y,Z⟧}
(k₁ k₂ : C⟦Z,W⟧)
(α : k₁ ==> k₂)
: rassociator _ _ _ o (α ▻ g ∘ f) = (α ▻ g) ▻ f o rassociator _ _ _.
Show proof.
Definition whisker_l_natural
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦X,Y⟧)
(α : k₁ ==> k₂)
: k₂ ◅ η o linvunitor k₂ o α = α ▻ f o (k₁ ◅ η) o linvunitor k₁.
Show proof.
Definition whisker_r_natural
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦Y,X⟧)
(α : k₁ ==> k₂)
: η ▻ k₂ o rinvunitor k₂ o α = (f ◅ α) o (η ▻ k₁) o rinvunitor k₁.
Show proof.
Definition whisker_l_iso_id₁
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦Y,X⟧)
(α : k₁ ==> k₂)
(inv_η : is_invertible_2cell η)
: α = runitor k₂ o (inv_η^-1 ▻ k₂) o (f ◅ α) o (η ▻ k₁) o rinvunitor k₁.
Show proof.
Definition whisker_r_iso_id₁
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦X,Y⟧)
(α : k₁ ==> k₂)
(inv_η : is_invertible_2cell η)
: α = lunitor k₂ o (k₂ ◅ inv_η^-1) o (α ▻ f) o (k₁ ◅ η) o linvunitor k₁.
Show proof.
Definition whisker_l_eq
{W X Y Z : C}
{f : C⟦X,Y⟧} {g : C⟦Y,Z⟧}
(k₁ k₂ : C⟦W,X⟧)
(α β : k₁ ==> k₂)
: f ◅ α = f ◅ β -> (g ∘ f) ◅ α = (g ∘ f) ◅ β.
Show proof.
Definition whisker_r_eq
{W X Y Z : C}
{f : C⟦Y,Z⟧} {g : C⟦X,Y⟧}
(k₁ k₂ : C⟦Z,W⟧)
(α β : k₁ ==> k₂)
: α ▻ f = β ▻ f -> α ▻ (f ∘ g) = β ▻ (f ∘ g).
Show proof.
Definition left_unit_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: (runitor g) ▻ f = runitor (g ∘ f) o lassociator f g (id₁ Z).
Show proof.
Definition left_unit_inv_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: (rinvunitor g) ▻ f = rassociator _ _ _ o rinvunitor (g ∘ f).
Show proof.
Definition lunitor_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: lunitor (g ∘ f) = g ◅ (lunitor f) o lassociator (id₁ X) f g.
Show proof.
Definition linvunitor_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: linvunitor (g ∘ f) = rassociator (id₁ X) f g o (g ◅ (linvunitor f)).
Show proof.
Definition lunitor_id_is_left_unit_id
(X : C)
: lunitor (id₁ X) = runitor (id₁ X).
Show proof.
Definition lunitor_V_id_is_left_unit_V_id
(X : C)
: linvunitor (id₁ X) = rinvunitor (id₁ X).
Show proof.
Definition left_unit_inv_assoc₂
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: rinvunitor (g ∘ f) = lassociator f g (id₁ Z) o (rinvunitor g ▻ f).
Show proof.
Definition triangle_l_inv
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: lassociator f (id₁ Y) g o linvunitor g ⋆⋆ id₂ f = id₂ g ⋆⋆ rinvunitor f.
Show proof.
End laws.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Bicat. Import Notations.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Invertible_2cells.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Unitors.
Local Open Scope bicategory_scope.
Local Open Scope cat.
Section laws.
Context {C : bicat}.
Definition triangle_r
{X Y Z : C}
(g : C⟦Y,Z⟧)
(f : C⟦X,Y⟧)
: lunitor g ⋆⋆ id₂ f = (id₂ g ⋆⋆ runitor f) o lassociator f (id₁ Y) g.
Show proof.
cbn.
apply pathsinv0.
unfold hcomp.
etrans.
{ apply maponpaths.
etrans. { apply maponpaths.
apply lwhisker_id2. }
apply id2_right. }
etrans. apply runitor_rwhisker.
apply pathsinv0.
etrans. { apply maponpaths_2. apply id2_rwhisker. }
apply id2_left.
apply pathsinv0.
unfold hcomp.
etrans.
{ apply maponpaths.
etrans. { apply maponpaths.
apply lwhisker_id2. }
apply id2_right. }
etrans. apply runitor_rwhisker.
apply pathsinv0.
etrans. { apply maponpaths_2. apply id2_rwhisker. }
apply id2_left.
Definition interchange
{X Y Z : C}
{f₁ g₁ h₁ : C⟦Y,Z⟧}
{f₂ g₂ h₂ : C⟦X,Y⟧}
(η₁ : f₁ ==> g₁) (η₂ : f₂ ==> g₂)
(ε₁ : g₁ ==> h₁) (ε₂ : g₂ ==> h₂)
: (ε₁ o η₁) ⋆⋆ (ε₂ o η₂) = (ε₁ ⋆⋆ ε₂) o (η₁ ⋆⋆ η₂).
Show proof.
apply hcomp_vcomp.
Definition rinvunitor_natural
{X Y : C}
{f g : C⟦X, Y⟧}
(η : f ==> g)
: rinvunitor g o η = (id₂ (id₁ Y) ⋆⋆ η) o rinvunitor f.
Show proof.
use (vcomp_rcancel (runitor _ )).
{ apply is_invertible_2cell_runitor. }
rewrite vassocl.
rewrite rinvunitor_runitor.
use (vcomp_lcancel (runitor _ )).
{ apply is_invertible_2cell_runitor. }
repeat rewrite vassocr.
rewrite runitor_rinvunitor.
rewrite id2_left, id2_right.
apply (! runitor_natural _ _ _ _ _ ).
{ apply is_invertible_2cell_runitor. }
rewrite vassocl.
rewrite rinvunitor_runitor.
use (vcomp_lcancel (runitor _ )).
{ apply is_invertible_2cell_runitor. }
repeat rewrite vassocr.
rewrite runitor_rinvunitor.
rewrite id2_left, id2_right.
apply (! runitor_natural _ _ _ _ _ ).
Definition linvunitor_natural
{X Y : C}
{f g : C⟦X, Y⟧}
(η : f ==> g)
: linvunitor g o η = (η ⋆⋆ id₂ (id₁ X)) o linvunitor f.
Show proof.
use (vcomp_rcancel (lunitor _ )).
{ apply is_invertible_2cell_lunitor. }
rewrite vassocl.
rewrite linvunitor_lunitor.
use (vcomp_lcancel (lunitor _ )).
{ apply is_invertible_2cell_lunitor. }
repeat rewrite vassocr.
rewrite lunitor_linvunitor.
rewrite id2_left, id2_right.
apply (! lunitor_natural _ _ _ _ _ ).
{ apply is_invertible_2cell_lunitor. }
rewrite vassocl.
rewrite linvunitor_lunitor.
use (vcomp_lcancel (lunitor _ )).
{ apply is_invertible_2cell_lunitor. }
repeat rewrite vassocr.
rewrite lunitor_linvunitor.
rewrite id2_left, id2_right.
apply (! lunitor_natural _ _ _ _ _ ).
Definition lwhisker_hcomp
{X Y Z : C}
{f g : C⟦Y,Z⟧}
(h : C⟦X, Y⟧)
(α : f ==> g)
: h ◃ α = id₂ h ⋆ α.
Show proof.
unfold hcomp.
rewrite id2_rwhisker.
rewrite id2_left.
reflexivity.
rewrite id2_rwhisker.
rewrite id2_left.
reflexivity.
Definition rwhisker_hcomp
{X Y Z : C}
{f g : C⟦X,Y⟧}
(h : C⟦Y,Z⟧)
(α : f ==> g)
: α ▹ h = α ⋆ id₂ h.
Show proof.
unfold hcomp.
rewrite lwhisker_id2.
rewrite id2_right.
reflexivity.
rewrite lwhisker_id2.
rewrite id2_right.
reflexivity.
Definition inverse_pentagon
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator f g (k ∘ h) o rassociator (g ∘ f) h k
=
(id₂ f ⋆ rassociator g h k) o (rassociator f (h ∘ g) k)
o (rassociator f g h ⋆ id₂ k).
Show proof.
use inv_cell_eq.
- is_iso.
- is_iso.
- cbn. rewrite <- !vassocr. apply pentagon.
- is_iso.
- is_iso.
- cbn. rewrite <- !vassocr. apply pentagon.
Definition inverse_pentagon_2
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator (g ∘ f) h k o (lassociator f g h ⋆ id2 k)
=
lassociator f g (k ∘ h) o (f ◃ rassociator g h k)
o rassociator f (h ∘ g) k.
Show proof.
rewrite <- !inverse_of_assoc.
use vcomp_move_R_Mp.
{
is_iso.
}
rewrite <- vassocr.
use vcomp_move_L_pM.
{
is_iso.
}
rewrite <- vassocr.
use vcomp_move_L_pM.
{
is_iso.
}
symmetry.
pose (pentagon k h g f) as p.
unfold hcomp in p.
rewrite id2_rwhisker in p.
rewrite id2_left in p.
exact p.
use vcomp_move_R_Mp.
{
is_iso.
}
rewrite <- vassocr.
use vcomp_move_L_pM.
{
is_iso.
}
rewrite <- vassocr.
use vcomp_move_L_pM.
{
is_iso.
}
symmetry.
pose (pentagon k h g f) as p.
unfold hcomp in p.
rewrite id2_rwhisker in p.
rewrite id2_left in p.
exact p.
Definition inverse_pentagon_3
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator f g (k ∘ h) o rassociator (g ∘ f) h k o (id₂ k ⋆⋆ lassociator f g h)
=
rassociator g h k ⋆⋆ id₂ f o rassociator f (h ∘ g) k.
Show proof.
use vcomp_move_R_pM.
{
is_iso.
}
cbn.
apply inverse_pentagon.
{
is_iso.
}
cbn.
apply inverse_pentagon.
Definition inverse_pentagon_4
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: (lassociator g h k ⋆⋆ id₂ f) o rassociator f g (k ∘ h)
=
rassociator f (h ∘ g) k o id₂ k ⋆⋆ rassociator f g h o lassociator (g ∘ f) h k.
Show proof.
rewrite <- !inverse_of_assoc.
use vcomp_move_R_pM.
{
is_iso.
}
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
symmetry ; apply pentagon.
use vcomp_move_R_pM.
{
is_iso.
}
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
symmetry ; apply pentagon.
Definition inverse_pentagon_5
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: lassociator f g (k ∘ h) o (rassociator g h k ⋆⋆ id₂ f)
=
rassociator (g ∘ f) h k o (id₂ k ⋆⋆ lassociator f g h) o lassociator f (h ∘ g) k.
Show proof.
rewrite <- !inverse_of_assoc.
use vcomp_move_R_pM.
{
is_iso.
}
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
apply pentagon.
use vcomp_move_R_pM.
{
is_iso.
}
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
apply pentagon.
Definition inverse_pentagon_6
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: rassociator f (h ∘ g) k o id₂ k ⋆⋆ rassociator f g h
=
lassociator g h k ⋆⋆ id₂ f o rassociator f g (k ∘ h) o rassociator (g ∘ f) h k.
Show proof.
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
cbn.
symmetry.
rewrite <- !vassocr.
apply inverse_pentagon.
use vcomp_move_L_Mp.
{
is_iso.
}
cbn.
symmetry.
rewrite <- !vassocr.
apply inverse_pentagon.
Definition pentagon_2
{V W X Y Z : C}
(k : C⟦Y,Z⟧) (h : C⟦X,Y⟧)
(g : C⟦W,X⟧) (f : C⟦V,W⟧)
: lassociator f (h ∘ g) k o lassociator g h k ⋆⋆ id₂ f
=
id₂ k ⋆⋆ rassociator f g h o lassociator (g ∘ f) h k o lassociator f g (k ∘ h).
Show proof.
rewrite <- !inverse_of_assoc.
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
symmetry ; apply pentagon.
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
symmetry ; apply pentagon.
Definition triangle_r_inv
{X Y Z : C}
(g : C ⟦ Y, Z ⟧) (f : C ⟦ X, Y ⟧)
: linvunitor g ⋆⋆ id₂ f
=
rassociator _ _ _ o id₂ g ⋆⋆ rinvunitor f.
Show proof.
use inv_cell_eq.
- is_iso.
- is_iso.
- cbn. apply triangle_r.
- is_iso.
- is_iso.
- cbn. apply triangle_r.
Definition triangle_l
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: lunitor g ⋆⋆ id₂ f o rassociator _ _ _ = id₂ g ⋆⋆ runitor f.
Show proof.
rewrite triangle_r.
rewrite vassocr.
rewrite <- inverse_of_assoc.
rewrite vcomp_lid.
rewrite id2_left.
reflexivity.
rewrite vassocr.
rewrite <- inverse_of_assoc.
rewrite vcomp_lid.
rewrite id2_left.
reflexivity.
Definition bc_whisker_r_compose
{X Y Z : C}
(f : C⟦X,Y⟧)
{g₁ g₂ g₃ : C⟦Y,Z⟧}
(p₁ : g₁ ==> g₂) (p₂ : g₂ ==> g₃)
: (p₂ o p₁) ▻ f = (p₂ ▻ f) o (p₁ ▻ f).
Show proof.
symmetry.
apply lwhisker_vcomp.
apply lwhisker_vcomp.
Definition bc_whisker_l_compose
{X Y Z : C}
{f₁ f₂ f₃ : C⟦X,Y⟧}
(g : C⟦Y,Z⟧)
(p₁ : f₁ ==> f₂) (p₂ : f₂ ==> f₃)
: g ◅ (p₂ o p₁) = (g ◅ p₂) o (g ◅ p₁).
Show proof.
symmetry.
apply rwhisker_vcomp.
apply rwhisker_vcomp.
Definition whisker_l_hcomp
{W X Y Z : C}
{f : C⟦X,Y⟧} {g : C⟦Y,Z⟧}
(k₁ k₂ : C⟦W,X⟧)
(α : k₁ ==> k₂)
: lassociator _ _ _ o (g ∘ f ◅ α) = g ◅ (f ◅ α) o lassociator _ _ _.
Show proof.
symmetry.
apply rwhisker_rwhisker.
apply rwhisker_rwhisker.
Definition whisker_r_hcomp
{W X Y Z : C}
{f : C⟦X,Y⟧} {g : C⟦Y,Z⟧}
(k₁ k₂ : C⟦Z,W⟧)
(α : k₁ ==> k₂)
: rassociator _ _ _ o (α ▻ g ∘ f) = (α ▻ g) ▻ f o rassociator _ _ _.
Show proof.
use vcomp_move_R_Mp.
{
is_iso.
}
cbn.
rewrite <- vassocr.
use vcomp_move_L_pM.
{
is_iso.
}
cbn.
symmetry.
apply @lwhisker_lwhisker.
{
is_iso.
}
cbn.
rewrite <- vassocr.
use vcomp_move_L_pM.
{
is_iso.
}
cbn.
symmetry.
apply @lwhisker_lwhisker.
Definition whisker_l_natural
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦X,Y⟧)
(α : k₁ ==> k₂)
: k₂ ◅ η o linvunitor k₂ o α = α ▻ f o (k₁ ◅ η) o linvunitor k₁.
Show proof.
rewrite lwhisker_hcomp, rwhisker_hcomp.
rewrite !vassocr.
rewrite linvunitor_natural.
rewrite <- !vassocr.
apply maponpaths.
rewrite rwhisker_hcomp.
rewrite <- !interchange.
rewrite !id2_right, !id2_left.
reflexivity.
rewrite !vassocr.
rewrite linvunitor_natural.
rewrite <- !vassocr.
apply maponpaths.
rewrite rwhisker_hcomp.
rewrite <- !interchange.
rewrite !id2_right, !id2_left.
reflexivity.
Definition whisker_r_natural
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦Y,X⟧)
(α : k₁ ==> k₂)
: η ▻ k₂ o rinvunitor k₂ o α = (f ◅ α) o (η ▻ k₁) o rinvunitor k₁.
Show proof.
rewrite lwhisker_hcomp, rwhisker_hcomp.
rewrite !vassocr.
rewrite rinvunitor_natural.
rewrite <- !vassocr.
apply maponpaths.
rewrite lwhisker_hcomp.
rewrite <- !interchange.
rewrite !id2_right, !id2_left.
reflexivity.
rewrite !vassocr.
rewrite rinvunitor_natural.
rewrite <- !vassocr.
apply maponpaths.
rewrite lwhisker_hcomp.
rewrite <- !interchange.
rewrite !id2_right, !id2_left.
reflexivity.
Definition whisker_l_iso_id₁
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦Y,X⟧)
(α : k₁ ==> k₂)
(inv_η : is_invertible_2cell η)
: α = runitor k₂ o (inv_η^-1 ▻ k₂) o (f ◅ α) o (η ▻ k₁) o rinvunitor k₁.
Show proof.
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
exact (whisker_r_natural η k₁ k₂ α).
use vcomp_move_L_Mp.
{
is_iso.
}
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
exact (whisker_r_natural η k₁ k₂ α).
Definition whisker_r_iso_id₁
{X Y : C}
{f : C⟦X,X⟧}
(η : id₁ X ==> f)
(k₁ k₂ : C⟦X,Y⟧)
(α : k₁ ==> k₂)
(inv_η : is_invertible_2cell η)
: α = lunitor k₂ o (k₂ ◅ inv_η^-1) o (α ▻ f) o (k₁ ◅ η) o linvunitor k₁.
Show proof.
rewrite !vassocr.
use vcomp_move_L_Mp.
{
is_iso.
}
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
exact (whisker_l_natural η k₁ k₂ α).
use vcomp_move_L_Mp.
{
is_iso.
}
use vcomp_move_L_Mp.
{
is_iso.
}
rewrite <- !vassocr.
exact (whisker_l_natural η k₁ k₂ α).
Definition whisker_l_eq
{W X Y Z : C}
{f : C⟦X,Y⟧} {g : C⟦Y,Z⟧}
(k₁ k₂ : C⟦W,X⟧)
(α β : k₁ ==> k₂)
: f ◅ α = f ◅ β -> (g ∘ f) ◅ α = (g ∘ f) ◅ β.
Show proof.
intros Hαβ.
rewrite !rwhisker_hcomp.
rewrite !rwhisker_hcomp in Hαβ.
rewrite <- !hcomp_identity.
apply (vcomp_rcancel (lassociator _ _ _)).
{
is_iso.
}
rewrite !hcomp_lassoc.
rewrite Hαβ.
reflexivity.
rewrite !rwhisker_hcomp.
rewrite !rwhisker_hcomp in Hαβ.
rewrite <- !hcomp_identity.
apply (vcomp_rcancel (lassociator _ _ _)).
{
is_iso.
}
rewrite !hcomp_lassoc.
rewrite Hαβ.
reflexivity.
Definition whisker_r_eq
{W X Y Z : C}
{f : C⟦Y,Z⟧} {g : C⟦X,Y⟧}
(k₁ k₂ : C⟦Z,W⟧)
(α β : k₁ ==> k₂)
: α ▻ f = β ▻ f -> α ▻ (f ∘ g) = β ▻ (f ∘ g).
Show proof.
intros Hαβ.
rewrite !lwhisker_hcomp.
rewrite !lwhisker_hcomp in Hαβ.
rewrite <- !hcomp_identity.
apply (vcomp_lcancel (lassociator _ _ _)).
{
is_iso.
}
rewrite <- !hcomp_lassoc.
rewrite Hαβ.
reflexivity.
rewrite !lwhisker_hcomp.
rewrite !lwhisker_hcomp in Hαβ.
rewrite <- !hcomp_identity.
apply (vcomp_lcancel (lassociator _ _ _)).
{
is_iso.
}
rewrite <- !hcomp_lassoc.
rewrite Hαβ.
reflexivity.
Definition left_unit_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: (runitor g) ▻ f = runitor (g ∘ f) o lassociator f g (id₁ Z).
Show proof.
rewrite <- runitor_triangle.
unfold assoc.
rewrite vassocr.
rewrite lassociator_rassociator.
rewrite id2_left.
reflexivity.
unfold assoc.
rewrite vassocr.
rewrite lassociator_rassociator.
rewrite id2_left.
reflexivity.
Definition left_unit_inv_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: (rinvunitor g) ▻ f = rassociator _ _ _ o rinvunitor (g ∘ f).
Show proof.
rewrite <- rinvunitor_triangle.
rewrite <- vassocr.
rewrite lassociator_rassociator.
rewrite id2_right.
reflexivity.
rewrite <- vassocr.
rewrite lassociator_rassociator.
rewrite id2_right.
reflexivity.
Definition lunitor_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: lunitor (g ∘ f) = g ◅ (lunitor f) o lassociator (id₁ X) f g.
Show proof.
symmetry.
apply lunitor_triangle.
apply lunitor_triangle.
Definition linvunitor_assoc
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: linvunitor (g ∘ f) = rassociator (id₁ X) f g o (g ◅ (linvunitor f)).
Show proof.
use vcomp_move_L_pM.
{
is_iso.
}
cbn.
use vcomp_move_R_Mp.
{
is_iso.
}
cbn. rewrite <- lunitor_triangle.
rewrite vassocr.
rewrite rassociator_lassociator.
rewrite id2_left.
reflexivity.
{
is_iso.
}
cbn.
use vcomp_move_R_Mp.
{
is_iso.
}
cbn. rewrite <- lunitor_triangle.
rewrite vassocr.
rewrite rassociator_lassociator.
rewrite id2_left.
reflexivity.
Definition lunitor_id_is_left_unit_id
(X : C)
: lunitor (id₁ X) = runitor (id₁ X).
Show proof.
apply lunitor_runitor_identity.
Definition lunitor_V_id_is_left_unit_V_id
(X : C)
: linvunitor (id₁ X) = rinvunitor (id₁ X).
Show proof.
use inv_cell_eq.
- is_iso.
- is_iso.
- cbn. apply lunitor_runitor_identity.
- is_iso.
- is_iso.
- cbn. apply lunitor_runitor_identity.
Definition left_unit_inv_assoc₂
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: rinvunitor (g ∘ f) = lassociator f g (id₁ Z) o (rinvunitor g ▻ f).
Show proof.
rewrite left_unit_inv_assoc.
rewrite <- !vassocr.
rewrite rassociator_lassociator.
rewrite id2_right.
reflexivity.
rewrite <- !vassocr.
rewrite rassociator_lassociator.
rewrite id2_right.
reflexivity.
Definition triangle_l_inv
{X Y Z : C}
(g : C⟦Y,Z⟧) (f : C⟦X,Y⟧)
: lassociator f (id₁ Y) g o linvunitor g ⋆⋆ id₂ f = id₂ g ⋆⋆ rinvunitor f.
Show proof.
use inv_cell_eq.
- is_iso.
- is_iso.
- cbn. apply triangle_l.
- is_iso.
- is_iso.
- cbn. apply triangle_l.
End laws.