Library UniMath.Bicategories.DisplayedBicats.DispTransportLaws
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.DisplayedCats.Core.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor.
Local Open Scope cat.
Definition transportf_disp_2cell
{B : bicat}
{D : disp_bicat B}
{x y : B}
{f g : x --> y}
{xx : D x}
{yy : D y}
(ff : xx -->[ f ] yy)
(p : f = g)
: disp_invertible_2cell (idtoiso_2_1 _ _ p) ff (transportf _ p ff).
Show proof.
Definition disp_psfunctor_mor_transportb
{B₁ B₂ : bicat}
{F : psfunctor B₁ B₂}
{D₁ : disp_bicat B₁}
{D₂ : disp_bicat B₂}
(FF : disp_psfunctor D₁ D₂ F)
{x y : B₁}
{f g : x --> y}
(p : f = g)
{xx : D₁ x}
{yy : D₁ y}
(gg : xx -->[ g ] yy)
: disp_psfunctor_mor
_ _ _ FF
(transportb (λ z, xx -->[ z ] yy) p gg)
=
transportb
(λ z, _ -->[ z ] _)
(maponpaths (#F) p)
(disp_psfunctor_mor _ _ _ FF gg).
Show proof.
Definition transportf_prewhisker_1cell
{B : bicat}
{D : disp_bicat B}
{x y z : B}
{xx : D x}
{yy : D y}
{zz : D z}
{f₁ f₂ : x --> y}
{g : y --> z}
(p : f₁ = f₂)
(ff : xx -->[ f₁ ] yy)
(gg : yy -->[ g ] zz)
: (transportf
(λ z, xx -->[ z ] yy)
p
ff
;;
gg
=
transportf
(λ z, xx -->[ z ] zz)
(maponpaths (λ z, z · _) p)
(ff ;; gg))%mor_disp.
Show proof.
Definition transportf_postwhisker_1cell
{B : bicat}
{D : disp_bicat B}
{x y z : B}
{xx : D x}
{yy : D y}
{zz : D z}
{f : x --> y}
{g₁ g₂ : y --> z}
(p : g₁ = g₂)
(ff : xx -->[ f ] yy)
(gg : yy -->[ g₁ ] zz)
: (ff
;;
transportf
(λ z, yy -->[ z ] zz)
p
gg
=
transportf
(λ z, xx -->[ z ] zz)
(maponpaths (λ z, _ · z) p)
(ff ;; gg))%mor_disp.
Show proof.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor.
Local Open Scope cat.
Definition transportf_disp_2cell
{B : bicat}
{D : disp_bicat B}
{x y : B}
{f g : x --> y}
{xx : D x}
{yy : D y}
(ff : xx -->[ f ] yy)
(p : f = g)
: disp_invertible_2cell (idtoiso_2_1 _ _ p) ff (transportf _ p ff).
Show proof.
Definition disp_psfunctor_mor_transportb
{B₁ B₂ : bicat}
{F : psfunctor B₁ B₂}
{D₁ : disp_bicat B₁}
{D₂ : disp_bicat B₂}
(FF : disp_psfunctor D₁ D₂ F)
{x y : B₁}
{f g : x --> y}
(p : f = g)
{xx : D₁ x}
{yy : D₁ y}
(gg : xx -->[ g ] yy)
: disp_psfunctor_mor
_ _ _ FF
(transportb (λ z, xx -->[ z ] yy) p gg)
=
transportb
(λ z, _ -->[ z ] _)
(maponpaths (#F) p)
(disp_psfunctor_mor _ _ _ FF gg).
Show proof.
Definition transportf_prewhisker_1cell
{B : bicat}
{D : disp_bicat B}
{x y z : B}
{xx : D x}
{yy : D y}
{zz : D z}
{f₁ f₂ : x --> y}
{g : y --> z}
(p : f₁ = f₂)
(ff : xx -->[ f₁ ] yy)
(gg : yy -->[ g ] zz)
: (transportf
(λ z, xx -->[ z ] yy)
p
ff
;;
gg
=
transportf
(λ z, xx -->[ z ] zz)
(maponpaths (λ z, z · _) p)
(ff ;; gg))%mor_disp.
Show proof.
Definition transportf_postwhisker_1cell
{B : bicat}
{D : disp_bicat B}
{x y z : B}
{xx : D x}
{yy : D y}
{zz : D z}
{f : x --> y}
{g₁ g₂ : y --> z}
(p : g₁ = g₂)
(ff : xx -->[ f ] yy)
(gg : yy -->[ g₁ ] zz)
: (ff
;;
transportf
(λ z, yy -->[ z ] zz)
p
gg
=
transportf
(λ z, xx -->[ z ] zz)
(maponpaths (λ z, _ · z) p)
(ff ;; gg))%mor_disp.
Show proof.