Library UniMath.Bicategories.DisplayedBicats.Examples.Trivial
Trivial display.
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Examples.Initial.
Require Import UniMath.Bicategories.Core.Examples.Final.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.Core.AdjointUnique.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions.
Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Local Open Scope cat.
Local Open Scope mor_disp_scope.
Definition transportf_trivial (A B : UU) (a b : A) (p : a = b) (x : B) :
x = transportf (λ x : A, B) p x.
Show proof.
Section Trivial_Displayed.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Examples.Initial.
Require Import UniMath.Bicategories.Core.Examples.Final.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.Core.AdjointUnique.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions.
Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Local Open Scope cat.
Local Open Scope mor_disp_scope.
Definition transportf_trivial (A B : UU) (a b : A) (p : a = b) (x : B) :
x = transportf (λ x : A, B) p x.
Show proof.
Section Trivial_Displayed.
Variable (B C : bicat).
Definition trivial_disp_cat_ob_mor : disp_cat_ob_mor B
:= make_disp_cat_ob_mor
B
(λ _ : B, C)
(λ (_ _ : B) (a b : C) _, C⟦a,b⟧).
Definition trivial_disp_cat_id_comp
: disp_cat_id_comp B trivial_disp_cat_ob_mor
:= (λ (_ : B) (a : C), identity a),,
(λ (_ _ _ : B) _ _ (a b c : C) (f : C⟦a,b⟧) (g : C⟦b,c⟧), f · g).
Definition trivial_disp_cat_data : disp_cat_data B
:= trivial_disp_cat_ob_mor,, trivial_disp_cat_id_comp.
Definition trivial_disp_prebicat_1_id_comp_cells : disp_prebicat_1_id_comp_cells B
:= tpair (λ C:disp_cat_data B, disp_2cell_struct C)
trivial_disp_cat_data
(λ _ _ _ _ _ a b f g, f ==> g).
Definition trivial_displayed_data : disp_prebicat_data B.
use (trivial_disp_prebicat_1_id_comp_cells,, _).
repeat apply make_dirprod; cbn.
- intros _ _ _. exact (λ a b f, id2 f).
- intros _ _ _. exact (λ a b f, lunitor f).
- intros _ _ _. exact (λ a b f, runitor f).
- intros _ _ _. exact (λ a b f, linvunitor f).
- intros _ _ _. exact (λ a b f, rinvunitor f).
- intros _ _ _ _ _ _ _. exact (λ a b c d f g h, rassociator f g h).
- intros _ _ _ _ _ _ _. exact (λ a b c d f g h, lassociator f g h).
- intros _ _ _ _ _ _ _. exact (λ a b f g h x y, vcomp2 x y).
- intros _ _ _ _ _ _ _. exact (λ a b c f g1 g2 x, lwhisker f x).
- intros _ _ _ _ _ _ _. exact (λ a b c f1 f2 g x, rwhisker g x).
Defined.
Lemma trivial_disp_prebicat_laws : disp_prebicat_laws trivial_displayed_data.
Show proof.
Definition trivial_displayed_prebicat : disp_prebicat B
:= trivial_displayed_data,, trivial_disp_prebicat_laws.
Definition trivial_displayed_bicat : disp_bicat B.
Show proof.
End Trivial_Displayed.
Definition prod_bicat
(B C : bicat)
: bicat
:= total_bicat (trivial_displayed_bicat B C).
Definition pairobj
{B C : bicat}
(X : B) (Y : C)
: prod_bicat B C
:= X ,, Y.
Definition pairmor
{B C : bicat}
{X₁ X₂ : B} {Y₁ Y₂ : C}
(f : X₁ --> X₂) (g : Y₁ --> Y₂)
: pairobj X₁ Y₁ --> pairobj X₂ Y₂
:= f ,, g.
Definition paircell
{B C : bicat}
{X₁ X₂ : B} {Y₁ Y₂ : C}
{f₁ f₂ : X₁ --> X₂} {g₁ g₂ : Y₁ --> Y₂}
(α : f₁ ==> f₂) (β : g₁ ==> g₂)
: pairmor f₁ g₁ ==> pairmor f₂ g₂
:= α ,, β.
Definition trivial_is_invertible_2cell_to_is_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f g : B ⟦ b₁, b₂ ⟧}
{α : f ==> g}
(Hα : is_invertible_2cell α)
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff : c₁ -->[ f ] c₂}
{gg : c₁ -->[ g ] c₂}
(β : ff ==> gg)
(Hβ : is_invertible_2cell β)
: is_disp_invertible_2cell Hα β.
Show proof.
Definition trivial_invertible_2cell_to_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
: invertible_2cell ff gg → disp_invertible_2cell (id2_invertible_2cell f) ff gg.
Show proof.
Definition trivial_is_disp_invertible_to_is_invertible_2cell
{B C : bicat}
{b₁ b₂ : B}
{f g : B ⟦ b₁, b₂ ⟧}
{α : f ==> g}
(Hα : is_invertible_2cell α)
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
(β : ff ==> gg)
(Hβ : is_disp_invertible_2cell Hα β)
: is_invertible_2cell β.
Show proof.
Definition trivial_disp_invertible_to_invertible_2cell
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
: disp_invertible_2cell (id2_invertible_2cell f) ff gg → invertible_2cell ff gg.
Show proof.
Definition trivial_invertible_to_disp_invertible_to_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
(α : invertible_2cell ff gg)
: trivial_disp_invertible_to_invertible_2cell
(trivial_invertible_2cell_to_disp_invertible α)
=
α.
Show proof.
Definition trivial_disp_invertible_to_invertible_to_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
(α : disp_invertible_2cell (id2_invertible_2cell f) ff gg)
: trivial_invertible_2cell_to_disp_invertible
(trivial_disp_invertible_to_invertible_2cell α)
=
α.
Show proof.
Definition trivial_invertible_2cell_weq_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
(ff gg : c₁ -->[ f] c₂)
: invertible_2cell ff gg ≃ disp_invertible_2cell (id2_invertible_2cell f) ff gg.
Show proof.
Definition trivial_is_univalent_2_1
{B C : bicat}
(HC : is_univalent_2_1 C)
: disp_univalent_2_1 (trivial_displayed_bicat B C).
Show proof.
Definition trivial_adj_equiv_to_disp_adj_equiv
{B C : bicat}
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
: adjoint_equivalence c₁ c₂
→
disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂.
Show proof.
Definition trivial_disp_adj_equiv_to_adj_equiv
{B C : bicat}
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
: disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂
→
adjoint_equivalence c₁ c₂.
Show proof.
Definition trivial_adj_equiv_to_disp_to_adj
{B C : bicat}
(HC_2_1 : is_univalent_2_1 C)
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
(e : adjoint_equivalence c₁ c₂)
: trivial_disp_adj_equiv_to_adj_equiv
(trivial_adj_equiv_to_disp_adj_equiv e)
=
e.
Show proof.
Definition trivial_disp_adj_equiv_to_adj_to_disp
{B C : bicat}
(HB_2_1 : is_univalent_2_1 B)
(HC_2_1 : is_univalent_2_1 C)
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
(e : disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂)
: trivial_adj_equiv_to_disp_adj_equiv
(trivial_disp_adj_equiv_to_adj_equiv e)
=
e.
Show proof.
Definition trivial_adj_equiv_weq_disp_adj_equiv
{B C : bicat}
(HB_2_1 : is_univalent_2_1 B)
(HC_2_1 : is_univalent_2_1 C)
{b : B}
(c₁ c₂ : trivial_displayed_bicat B C b)
: adjoint_equivalence c₁ c₂
≃
disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂.
Show proof.
Definition trivial_is_univalent_2_0
{B C : bicat}
(HB_2_1 : is_univalent_2_1 B)
(HC_2_0 : is_univalent_2_0 C)
(HC_2_1 : is_univalent_2_1 C)
: disp_univalent_2_0 (trivial_displayed_bicat B C).
Show proof.
Definition trivial_disp_cat_ob_mor : disp_cat_ob_mor B
:= make_disp_cat_ob_mor
B
(λ _ : B, C)
(λ (_ _ : B) (a b : C) _, C⟦a,b⟧).
Definition trivial_disp_cat_id_comp
: disp_cat_id_comp B trivial_disp_cat_ob_mor
:= (λ (_ : B) (a : C), identity a),,
(λ (_ _ _ : B) _ _ (a b c : C) (f : C⟦a,b⟧) (g : C⟦b,c⟧), f · g).
Definition trivial_disp_cat_data : disp_cat_data B
:= trivial_disp_cat_ob_mor,, trivial_disp_cat_id_comp.
Definition trivial_disp_prebicat_1_id_comp_cells : disp_prebicat_1_id_comp_cells B
:= tpair (λ C:disp_cat_data B, disp_2cell_struct C)
trivial_disp_cat_data
(λ _ _ _ _ _ a b f g, f ==> g).
Definition trivial_displayed_data : disp_prebicat_data B.
use (trivial_disp_prebicat_1_id_comp_cells,, _).
repeat apply make_dirprod; cbn.
- intros _ _ _. exact (λ a b f, id2 f).
- intros _ _ _. exact (λ a b f, lunitor f).
- intros _ _ _. exact (λ a b f, runitor f).
- intros _ _ _. exact (λ a b f, linvunitor f).
- intros _ _ _. exact (λ a b f, rinvunitor f).
- intros _ _ _ _ _ _ _. exact (λ a b c d f g h, rassociator f g h).
- intros _ _ _ _ _ _ _. exact (λ a b c d f g h, lassociator f g h).
- intros _ _ _ _ _ _ _. exact (λ a b f g h x y, vcomp2 x y).
- intros _ _ _ _ _ _ _. exact (λ a b c f g1 g2 x, lwhisker f x).
- intros _ _ _ _ _ _ _. exact (λ a b c f1 f2 g x, rwhisker g x).
Defined.
Lemma trivial_disp_prebicat_laws : disp_prebicat_laws trivial_displayed_data.
Show proof.
repeat apply make_dirprod; red; cbn; intros.
- etrans. { apply id2_left. } apply transportf_trivial.
- etrans. { apply id2_right. } apply transportf_trivial.
- etrans. { apply vassocr. } apply transportf_trivial.
- etrans. { apply lwhisker_id2. } apply transportf_trivial.
- etrans. { apply id2_rwhisker. } apply transportf_trivial.
- etrans. { apply lwhisker_vcomp. } apply transportf_trivial.
- etrans. { apply rwhisker_vcomp. } apply transportf_trivial.
- etrans. { apply vcomp_lunitor. } apply transportf_trivial.
- etrans. { apply vcomp_runitor. } apply transportf_trivial.
- etrans. { apply lwhisker_lwhisker. } apply transportf_trivial.
- etrans. { apply rwhisker_lwhisker. } apply transportf_trivial.
- etrans. { apply rwhisker_rwhisker. } apply transportf_trivial.
- etrans. { apply vcomp_whisker. } apply transportf_trivial.
- etrans. { apply lunitor_linvunitor. } apply transportf_trivial.
- etrans. { apply linvunitor_lunitor. } apply transportf_trivial.
- etrans. { apply runitor_rinvunitor. } apply transportf_trivial.
- etrans. { apply rinvunitor_runitor. } apply transportf_trivial.
- etrans. { apply lassociator_rassociator. } apply transportf_trivial.
- etrans. { apply rassociator_lassociator. } apply transportf_trivial.
- etrans. { apply runitor_rwhisker. } apply transportf_trivial.
- etrans. { apply lassociator_lassociator. } apply transportf_trivial.
- etrans. { apply id2_left. } apply transportf_trivial.
- etrans. { apply id2_right. } apply transportf_trivial.
- etrans. { apply vassocr. } apply transportf_trivial.
- etrans. { apply lwhisker_id2. } apply transportf_trivial.
- etrans. { apply id2_rwhisker. } apply transportf_trivial.
- etrans. { apply lwhisker_vcomp. } apply transportf_trivial.
- etrans. { apply rwhisker_vcomp. } apply transportf_trivial.
- etrans. { apply vcomp_lunitor. } apply transportf_trivial.
- etrans. { apply vcomp_runitor. } apply transportf_trivial.
- etrans. { apply lwhisker_lwhisker. } apply transportf_trivial.
- etrans. { apply rwhisker_lwhisker. } apply transportf_trivial.
- etrans. { apply rwhisker_rwhisker. } apply transportf_trivial.
- etrans. { apply vcomp_whisker. } apply transportf_trivial.
- etrans. { apply lunitor_linvunitor. } apply transportf_trivial.
- etrans. { apply linvunitor_lunitor. } apply transportf_trivial.
- etrans. { apply runitor_rinvunitor. } apply transportf_trivial.
- etrans. { apply rinvunitor_runitor. } apply transportf_trivial.
- etrans. { apply lassociator_rassociator. } apply transportf_trivial.
- etrans. { apply rassociator_lassociator. } apply transportf_trivial.
- etrans. { apply runitor_rwhisker. } apply transportf_trivial.
- etrans. { apply lassociator_lassociator. } apply transportf_trivial.
Definition trivial_displayed_prebicat : disp_prebicat B
:= trivial_displayed_data,, trivial_disp_prebicat_laws.
Definition trivial_displayed_bicat : disp_bicat B.
Show proof.
End Trivial_Displayed.
Definition prod_bicat
(B C : bicat)
: bicat
:= total_bicat (trivial_displayed_bicat B C).
Definition pairobj
{B C : bicat}
(X : B) (Y : C)
: prod_bicat B C
:= X ,, Y.
Definition pairmor
{B C : bicat}
{X₁ X₂ : B} {Y₁ Y₂ : C}
(f : X₁ --> X₂) (g : Y₁ --> Y₂)
: pairobj X₁ Y₁ --> pairobj X₂ Y₂
:= f ,, g.
Definition paircell
{B C : bicat}
{X₁ X₂ : B} {Y₁ Y₂ : C}
{f₁ f₂ : X₁ --> X₂} {g₁ g₂ : Y₁ --> Y₂}
(α : f₁ ==> f₂) (β : g₁ ==> g₂)
: pairmor f₁ g₁ ==> pairmor f₂ g₂
:= α ,, β.
Definition trivial_is_invertible_2cell_to_is_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f g : B ⟦ b₁, b₂ ⟧}
{α : f ==> g}
(Hα : is_invertible_2cell α)
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff : c₁ -->[ f ] c₂}
{gg : c₁ -->[ g ] c₂}
(β : ff ==> gg)
(Hβ : is_invertible_2cell β)
: is_disp_invertible_2cell Hα β.
Show proof.
simple refine (_ ,, (_ ,, _)).
- exact (Hβ^-1).
- abstract
(unfold transportb ; cbn ;
rewrite transportf_const ;
apply vcomp_rinv).
- abstract
(unfold transportb ; cbn ;
rewrite transportf_const ;
apply vcomp_linv).
- exact (Hβ^-1).
- abstract
(unfold transportb ; cbn ;
rewrite transportf_const ;
apply vcomp_rinv).
- abstract
(unfold transportb ; cbn ;
rewrite transportf_const ;
apply vcomp_linv).
Definition trivial_invertible_2cell_to_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
: invertible_2cell ff gg → disp_invertible_2cell (id2_invertible_2cell f) ff gg.
Show proof.
intros α.
simple refine (_ ,, _).
- exact (pr1 α).
- apply trivial_is_invertible_2cell_to_is_disp_invertible.
exact (pr2 α).
simple refine (_ ,, _).
- exact (pr1 α).
- apply trivial_is_invertible_2cell_to_is_disp_invertible.
exact (pr2 α).
Definition trivial_is_disp_invertible_to_is_invertible_2cell
{B C : bicat}
{b₁ b₂ : B}
{f g : B ⟦ b₁, b₂ ⟧}
{α : f ==> g}
(Hα : is_invertible_2cell α)
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
(β : ff ==> gg)
(Hβ : is_disp_invertible_2cell Hα β)
: is_invertible_2cell β.
Show proof.
use make_is_invertible_2cell.
- exact (pr1 Hβ).
- abstract
(pose (pr12 Hβ) as p ;
cbn in p ; unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- abstract
(pose (pr22 Hβ) as p ;
cbn in p ; unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- exact (pr1 Hβ).
- abstract
(pose (pr12 Hβ) as p ;
cbn in p ; unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- abstract
(pose (pr22 Hβ) as p ;
cbn in p ; unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
Definition trivial_disp_invertible_to_invertible_2cell
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
: disp_invertible_2cell (id2_invertible_2cell f) ff gg → invertible_2cell ff gg.
Show proof.
intros α.
use make_invertible_2cell.
- exact (pr1 α).
- use (trivial_is_disp_invertible_to_is_invertible_2cell
(is_invertible_2cell_id₂ f)).
apply α.
use make_invertible_2cell.
- exact (pr1 α).
- use (trivial_is_disp_invertible_to_is_invertible_2cell
(is_invertible_2cell_id₂ f)).
apply α.
Definition trivial_invertible_to_disp_invertible_to_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
(α : invertible_2cell ff gg)
: trivial_disp_invertible_to_invertible_2cell
(trivial_invertible_2cell_to_disp_invertible α)
=
α.
Show proof.
Definition trivial_disp_invertible_to_invertible_to_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
{ff gg : c₁ -->[ f] c₂}
(α : disp_invertible_2cell (id2_invertible_2cell f) ff gg)
: trivial_invertible_2cell_to_disp_invertible
(trivial_disp_invertible_to_invertible_2cell α)
=
α.
Show proof.
Definition trivial_invertible_2cell_weq_disp_invertible
{B C : bicat}
{b₁ b₂ : B}
{f : B ⟦ b₁, b₂ ⟧}
{c₁ : trivial_displayed_bicat B C b₁}
{c₂ : trivial_displayed_bicat B C b₂}
(ff gg : c₁ -->[ f] c₂)
: invertible_2cell ff gg ≃ disp_invertible_2cell (id2_invertible_2cell f) ff gg.
Show proof.
use make_weq.
- exact trivial_invertible_2cell_to_disp_invertible.
- use isweq_iso.
+ exact trivial_disp_invertible_to_invertible_2cell.
+ exact trivial_invertible_to_disp_invertible_to_invertible.
+ exact trivial_disp_invertible_to_invertible_to_disp_invertible.
- exact trivial_invertible_2cell_to_disp_invertible.
- use isweq_iso.
+ exact trivial_disp_invertible_to_invertible_2cell.
+ exact trivial_invertible_to_disp_invertible_to_invertible.
+ exact trivial_disp_invertible_to_invertible_to_disp_invertible.
Definition trivial_is_univalent_2_1
{B C : bicat}
(HC : is_univalent_2_1 C)
: disp_univalent_2_1 (trivial_displayed_bicat B C).
Show proof.
use fiberwise_local_univalent_is_univalent_2_1.
intros b₁ b₂ f c₁ c₂ ff gg.
use weqhomot.
- exact (trivial_invertible_2cell_weq_disp_invertible ff gg
∘ make_weq (idtoiso_2_1 ff gg) (HC _ _ ff gg))%weq.
- intro p ; cbn in p.
induction p.
use subtypePath.
{
intro ; apply isaprop_is_disp_invertible_2cell.
}
apply idpath.
intros b₁ b₂ f c₁ c₂ ff gg.
use weqhomot.
- exact (trivial_invertible_2cell_weq_disp_invertible ff gg
∘ make_weq (idtoiso_2_1 ff gg) (HC _ _ ff gg))%weq.
- intro p ; cbn in p.
induction p.
use subtypePath.
{
intro ; apply isaprop_is_disp_invertible_2cell.
}
apply idpath.
Definition trivial_adj_equiv_to_disp_adj_equiv
{B C : bicat}
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
: adjoint_equivalence c₁ c₂
→
disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂.
Show proof.
intros e.
simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))).
- exact (pr1 e).
- exact (left_adjoint_right_adjoint e).
- exact (left_adjoint_unit e).
- exact (left_adjoint_counit e).
- abstract
(unfold transportb ;
cbn ;
rewrite transportf_const ; cbn ;
exact (pr1 (pr122 e))).
- abstract
(unfold transportb ;
cbn ;
rewrite transportf_const ; cbn ;
exact (pr2 (pr122 e))).
- abstract
(apply trivial_is_invertible_2cell_to_is_disp_invertible ;
apply (pr2 e)).
- abstract
(apply trivial_is_invertible_2cell_to_is_disp_invertible ;
apply (pr2 e)).
simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))).
- exact (pr1 e).
- exact (left_adjoint_right_adjoint e).
- exact (left_adjoint_unit e).
- exact (left_adjoint_counit e).
- abstract
(unfold transportb ;
cbn ;
rewrite transportf_const ; cbn ;
exact (pr1 (pr122 e))).
- abstract
(unfold transportb ;
cbn ;
rewrite transportf_const ; cbn ;
exact (pr2 (pr122 e))).
- abstract
(apply trivial_is_invertible_2cell_to_is_disp_invertible ;
apply (pr2 e)).
- abstract
(apply trivial_is_invertible_2cell_to_is_disp_invertible ;
apply (pr2 e)).
Definition trivial_disp_adj_equiv_to_adj_equiv
{B C : bicat}
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
: disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂
→
adjoint_equivalence c₁ c₂.
Show proof.
intros e.
simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))).
- exact (pr1 e).
- exact (pr112 e).
- exact (pr1 (pr212 e)).
- exact (pr2 (pr212 e)).
- abstract
(pose (pr1 (pr122 e)) as p ;
cbn in p ;
unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- abstract
(pose (pr2 (pr122 e)) as p ;
cbn in p ;
unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- abstract
(apply (trivial_is_disp_invertible_to_is_invertible_2cell _ _ (pr1 (pr222 e)))).
- abstract
(apply (trivial_is_disp_invertible_to_is_invertible_2cell _ _ (pr2 (pr222 e)))).
simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))).
- exact (pr1 e).
- exact (pr112 e).
- exact (pr1 (pr212 e)).
- exact (pr2 (pr212 e)).
- abstract
(pose (pr1 (pr122 e)) as p ;
cbn in p ;
unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- abstract
(pose (pr2 (pr122 e)) as p ;
cbn in p ;
unfold transportb in p ;
rewrite transportf_const in p ;
exact p).
- abstract
(apply (trivial_is_disp_invertible_to_is_invertible_2cell _ _ (pr1 (pr222 e)))).
- abstract
(apply (trivial_is_disp_invertible_to_is_invertible_2cell _ _ (pr2 (pr222 e)))).
Definition trivial_adj_equiv_to_disp_to_adj
{B C : bicat}
(HC_2_1 : is_univalent_2_1 C)
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
(e : adjoint_equivalence c₁ c₂)
: trivial_disp_adj_equiv_to_adj_equiv
(trivial_adj_equiv_to_disp_adj_equiv e)
=
e.
Show proof.
Definition trivial_disp_adj_equiv_to_adj_to_disp
{B C : bicat}
(HB_2_1 : is_univalent_2_1 B)
(HC_2_1 : is_univalent_2_1 C)
{b : B}
{c₁ c₂ : trivial_displayed_bicat B C b}
(e : disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂)
: trivial_adj_equiv_to_disp_adj_equiv
(trivial_disp_adj_equiv_to_adj_equiv e)
=
e.
Show proof.
use subtypePath.
{
intro.
use isaprop_disp_left_adjoint_equivalence.
- exact HB_2_1.
- apply trivial_is_univalent_2_1.
exact HC_2_1.
}
apply idpath.
{
intro.
use isaprop_disp_left_adjoint_equivalence.
- exact HB_2_1.
- apply trivial_is_univalent_2_1.
exact HC_2_1.
}
apply idpath.
Definition trivial_adj_equiv_weq_disp_adj_equiv
{B C : bicat}
(HB_2_1 : is_univalent_2_1 B)
(HC_2_1 : is_univalent_2_1 C)
{b : B}
(c₁ c₂ : trivial_displayed_bicat B C b)
: adjoint_equivalence c₁ c₂
≃
disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂.
Show proof.
use make_weq.
- exact trivial_adj_equiv_to_disp_adj_equiv.
- use isweq_iso.
+ exact trivial_disp_adj_equiv_to_adj_equiv.
+ exact (trivial_adj_equiv_to_disp_to_adj HC_2_1).
+ exact (trivial_disp_adj_equiv_to_adj_to_disp HB_2_1 HC_2_1).
- exact trivial_adj_equiv_to_disp_adj_equiv.
- use isweq_iso.
+ exact trivial_disp_adj_equiv_to_adj_equiv.
+ exact (trivial_adj_equiv_to_disp_to_adj HC_2_1).
+ exact (trivial_disp_adj_equiv_to_adj_to_disp HB_2_1 HC_2_1).
Definition trivial_is_univalent_2_0
{B C : bicat}
(HB_2_1 : is_univalent_2_1 B)
(HC_2_0 : is_univalent_2_0 C)
(HC_2_1 : is_univalent_2_1 C)
: disp_univalent_2_0 (trivial_displayed_bicat B C).
Show proof.
use fiberwise_univalent_2_0_to_disp_univalent_2_0.
intros b c₁ c₂.
use weqhomot.
- exact (trivial_adj_equiv_weq_disp_adj_equiv HB_2_1 HC_2_1 c₁ c₂
∘ make_weq (idtoiso_2_0 c₁ c₂) (HC_2_0 c₁ c₂))%weq.
- intros p.
cbn in p.
induction p.
use subtypePath.
{
intro.
use isaprop_disp_left_adjoint_equivalence.
- exact HB_2_1.
- apply trivial_is_univalent_2_1.
exact HC_2_1.
}
apply idpath.
intros b c₁ c₂.
use weqhomot.
- exact (trivial_adj_equiv_weq_disp_adj_equiv HB_2_1 HC_2_1 c₁ c₂
∘ make_weq (idtoiso_2_0 c₁ c₂) (HC_2_0 c₁ c₂))%weq.
- intros p.
cbn in p.
induction p.
use subtypePath.
{
intro.
use isaprop_disp_left_adjoint_equivalence.
- exact HB_2_1.
- apply trivial_is_univalent_2_1.
exact HC_2_1.
}
apply idpath.