Library UniMath.Bicategories.DisplayedBicats.ExamplesOfCleavings.TrivialCleaving
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.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.EquivToAdjequiv.
Require Import UniMath.Bicategories.Morphisms.FullyFaithful.
Require Import UniMath.Bicategories.Morphisms.Properties.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations.
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.DisplayedBicats.CleavingOfBicat.
Require Import UniMath.Bicategories.DisplayedBicats.Cartesians.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.Trivial.
Local Open Scope cat.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.EquivToAdjequiv.
Require Import UniMath.Bicategories.Morphisms.FullyFaithful.
Require Import UniMath.Bicategories.Morphisms.Properties.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations.
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.DisplayedBicats.CleavingOfBicat.
Require Import UniMath.Bicategories.DisplayedBicats.Cartesians.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.Trivial.
Local Open Scope cat.
The trivial bicategory has a cleaving
Characterisation of cartesian 2-cells
Definition trivial_invertible_is_cartesian_2cell
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_invertible_2cell β)
: is_cartesian_2cell (trivial_displayed_bicat B₁ B₂) β.
Show proof.
Definition trivial_cartesian_2cell_is_invertible
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_cartesian_2cell (trivial_displayed_bicat B₁ B₂) β)
: is_invertible_2cell β.
Show proof.
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_invertible_2cell β)
: is_cartesian_2cell (trivial_displayed_bicat B₁ B₂) β.
Show proof.
intros h h' γ ββ ; cbn in *.
use iscontraprop1.
- abstract
(use invproofirrelevance ;
intros τ₁ τ₂ ;
use subtypePath ; [ intro ; apply cellset_property | ] ;
use (vcomp_rcancel _ Hβ) ;
exact (pr2 τ₁ @ !(pr2 τ₂))).
- refine (ββ • Hβ^-1 ,, _).
abstract
(rewrite !vassocl ;
rewrite vcomp_linv ;
apply id2_right).
use iscontraprop1.
- abstract
(use invproofirrelevance ;
intros τ₁ τ₂ ;
use subtypePath ; [ intro ; apply cellset_property | ] ;
use (vcomp_rcancel _ Hβ) ;
exact (pr2 τ₁ @ !(pr2 τ₂))).
- refine (ββ • Hβ^-1 ,, _).
abstract
(rewrite !vassocl ;
rewrite vcomp_linv ;
apply id2_right).
Definition trivial_cartesian_2cell_is_invertible
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_cartesian_2cell (trivial_displayed_bicat B₁ B₂) β)
: is_invertible_2cell β.
Show proof.
cbn in *.
unfold is_cartesian_2cell in Hβ.
pose (Hβ f₁ g₂ (id2 _) (id2 _)) as lift.
pose (inv := iscontrpr1 lift).
cbn in lift, inv.
use make_is_invertible_2cell.
- exact (pr1 inv).
- abstract
(pose (Hβ f₁ g₁ (id2 _) β) as m ;
cbn in m ;
pose (proofirrelevance _ (isapropifcontr m)) as i ;
simple refine (maponpaths pr1 (i (β • pr1 inv ,, _) (id₂ g₁ ,, _))) ; cbn ;
[ cbn ;
rewrite vassocl ;
rewrite (pr2 inv) ;
apply id2_right
| apply id2_left ]).
- abstract
(exact (pr2 inv)).
unfold is_cartesian_2cell in Hβ.
pose (Hβ f₁ g₂ (id2 _) (id2 _)) as lift.
pose (inv := iscontrpr1 lift).
cbn in lift, inv.
use make_is_invertible_2cell.
- exact (pr1 inv).
- abstract
(pose (Hβ f₁ g₁ (id2 _) β) as m ;
cbn in m ;
pose (proofirrelevance _ (isapropifcontr m)) as i ;
simple refine (maponpaths pr1 (i (β • pr1 inv ,, _) (id₂ g₁ ,, _))) ; cbn ;
[ cbn ;
rewrite vassocl ;
rewrite (pr2 inv) ;
apply id2_right
| apply id2_left ]).
- abstract
(exact (pr2 inv)).
Characterisation of cartesian 2-cells
Definition trivial_invertible_is_opcartesian_2cell
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_invertible_2cell β)
: is_opcartesian_2cell (trivial_displayed_bicat B₁ B₂) β.
Show proof.
Definition trivial_opcartesian_2cell_is_invertible
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_opcartesian_2cell (trivial_displayed_bicat B₁ B₂) β)
: is_invertible_2cell β.
Show proof.
Definition trivial_local_cleaving
: local_cleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_local_opcleaving
: local_opcleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_local_isocleaving
: local_iso_cleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
Section TrivialCartesianOneCell.
Context {x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
(f : x₁ --> x₂)
(g : y₁ -->[ f ] y₂)
(Hg : left_adjoint_equivalence g).
Definition trivial_lift_1cell
{x₃ : B₁}
{y₃ : trivial_displayed_bicat B₁ B₂ x₃}
(h : x₃ --> x₁)
(k : y₃ -->[ h · f] y₂)
: lift_1cell_factor (trivial_displayed_bicat B₁ B₂) g k.
Show proof.
Definition trivial_lift_2cell
{w : B₁}
{z : B₂}
{h h' : w --> x₁}
{k k' : z --> y₂}
(γ : h ==> h')
(δ : k ==> k')
(Lh : lift_1cell_factor (trivial_displayed_bicat B₁ B₂) g k)
(Lh' : lift_1cell_factor (trivial_displayed_bicat B₁ B₂) g k')
: lift_2cell_factor (trivial_displayed_bicat B₁ B₂) g (δ := γ) δ Lh Lh'.
Show proof.
Definition trivial_cartesian_1cell
: cartesian_1cell (trivial_displayed_bicat B₁ B₂) g.
Show proof.
End TrivialCartesianOneCell.
Section Cartesian1CellToAdjEquiv.
Context {x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
(f : x₁ --> x₂)
(g : y₁ -->[ f ] y₂)
(Hg : cartesian_1cell (trivial_displayed_bicat B₁ B₂) g).
Let inv : y₂ --> y₁ := pr1 (pr1 Hg x₁ y₂ (id₁ _) (id₁ _)).
Let ε : inv · g ==> id₁ y₂ := pr12 (pr1 Hg x₁ y₂ (id₁ _) (id₁ _)).
Let εinv
: is_invertible_2cell ε
:= trivial_disp_invertible_to_invertible_2cell
(pr2 (pr1 Hg x₁ y₂ (id₁ _) (id₁ _))).
Local Definition unit_help_lift₁
: lift_1cell_factor (trivial_displayed_bicat B₁ B₂) (h := id₁ _) g g.
Show proof.
Local Definition unit_help_lift₂
: lift_1cell_factor (trivial_displayed_bicat B₁ B₂) (h := id₁ _) g g.
Show proof.
Local Definition unit
: id₁ y₁ ==> g · inv
:= cartesian_1cell_lift_2cell
_ _
Hg
(δ := id2 _) (id2 _)
unit_help_lift₁ unit_help_lift₂.
Definition trivial_cartesian_1cell_left_adj_data
: left_adjoint_data g
:= (inv ,, unit ,, ε).
Definition trivial_cartesian_1cell_left_equiv_axioms
: left_equivalence_axioms trivial_cartesian_1cell_left_adj_data.
Show proof.
Definition trivial_cartesian_1cell_is_adj_equiv
: left_adjoint_equivalence g.
Show proof.
Definition trivial_global_cleaving
: global_cleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_lwhisker_cartesian
: lwhisker_cartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_rwhisker_cartesian
: rwhisker_cartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_lwhisker_opcartesian
: lwhisker_opcartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_rwhisker_opcartesian
: rwhisker_opcartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
Definition trivial_cleaving_of_bicats
: cleaving_of_bicats (trivial_displayed_bicat B₁ B₂).
Show proof.
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_invertible_2cell β)
: is_opcartesian_2cell (trivial_displayed_bicat B₁ B₂) β.
Show proof.
intros h h' γ ββ ; cbn in *.
use iscontraprop1.
- abstract
(use invproofirrelevance ;
intros τ₁ τ₂ ;
use subtypePath ; [ intro ; apply cellset_property | ] ;
use (vcomp_lcancel _ Hβ) ;
exact (pr2 τ₁ @ !(pr2 τ₂))).
- refine (Hβ^-1 • ββ,, _).
abstract
(rewrite !vassocr ;
rewrite vcomp_rinv ;
apply id2_left).
use iscontraprop1.
- abstract
(use invproofirrelevance ;
intros τ₁ τ₂ ;
use subtypePath ; [ intro ; apply cellset_property | ] ;
use (vcomp_lcancel _ Hβ) ;
exact (pr2 τ₁ @ !(pr2 τ₂))).
- refine (Hβ^-1 • ββ,, _).
abstract
(rewrite !vassocr ;
rewrite vcomp_rinv ;
apply id2_left).
Definition trivial_opcartesian_2cell_is_invertible
{x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
{f₁ f₂ : x₁ --> x₂}
{g₁ : y₁ -->[ f₁ ] y₂}
{g₂ : y₁ -->[ f₂ ] y₂}
(α : f₁ ==> f₂)
(β : g₁ ==>[ α ] g₂)
(Hβ : is_opcartesian_2cell (trivial_displayed_bicat B₁ B₂) β)
: is_invertible_2cell β.
Show proof.
cbn in *.
unfold is_cartesian_2cell in Hβ.
pose (Hβ f₂ g₁ (id2 _) (id2 _)) as lift.
pose (inv := iscontrpr1 lift).
cbn in lift, inv.
use make_is_invertible_2cell.
- exact (pr1 inv).
- abstract
(exact (pr2 inv)).
- abstract
(pose (Hβ f₂ g₂ (id2 _) β) as m ;
cbn in m ;
pose (proofirrelevance _ (isapropifcontr m)) as i ;
simple refine (maponpaths pr1 (i (pr1 inv • β,, _) (id₂ g₂ ,, _))) ; cbn ;
[ cbn ;
rewrite vassocr ;
rewrite (pr2 inv) ;
apply id2_left
| apply id2_right ]).
unfold is_cartesian_2cell in Hβ.
pose (Hβ f₂ g₁ (id2 _) (id2 _)) as lift.
pose (inv := iscontrpr1 lift).
cbn in lift, inv.
use make_is_invertible_2cell.
- exact (pr1 inv).
- abstract
(exact (pr2 inv)).
- abstract
(pose (Hβ f₂ g₂ (id2 _) β) as m ;
cbn in m ;
pose (proofirrelevance _ (isapropifcontr m)) as i ;
simple refine (maponpaths pr1 (i (pr1 inv • β,, _) (id₂ g₂ ,, _))) ; cbn ;
[ cbn ;
rewrite vassocr ;
rewrite (pr2 inv) ;
apply id2_left
| apply id2_right ]).
Definition trivial_local_cleaving
: local_cleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
intros x₁ x₂ y₁ y₂ f₁ f₂ f α.
simple refine (f ,, id2 _ ,, _) ; cbn.
apply trivial_invertible_is_cartesian_2cell.
is_iso.
simple refine (f ,, id2 _ ,, _) ; cbn.
apply trivial_invertible_is_cartesian_2cell.
is_iso.
Definition trivial_local_opcleaving
: local_opcleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
intros x₁ x₂ y₁ y₂ f₁ f₂ f α.
simple refine (f ,, id2 _ ,, _) ; cbn.
apply trivial_invertible_is_opcartesian_2cell.
is_iso.
simple refine (f ,, id2 _ ,, _) ; cbn.
apply trivial_invertible_is_opcartesian_2cell.
is_iso.
Definition trivial_local_isocleaving
: local_iso_cleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
intros x₁ x₂ y₁ y₂ f₁ f₂ f α.
refine (f ,, id2 _ ,, _).
apply trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
refine (f ,, id2 _ ,, _).
apply trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
Section TrivialCartesianOneCell.
Context {x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
(f : x₁ --> x₂)
(g : y₁ -->[ f ] y₂)
(Hg : left_adjoint_equivalence g).
Definition trivial_lift_1cell
{x₃ : B₁}
{y₃ : trivial_displayed_bicat B₁ B₂ x₃}
(h : x₃ --> x₁)
(k : y₃ -->[ h · f] y₂)
: lift_1cell_factor (trivial_displayed_bicat B₁ B₂) g k.
Show proof.
simple refine (_ ,, _) ; cbn in *.
- exact (k · left_adjoint_right_adjoint Hg).
- simple refine (_ ,, _) ; cbn.
+ refine (rassociator _ _ _ • (k ◃ _) • runitor _).
exact (left_adjoint_counit Hg).
+ cbn.
use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
apply (left_equivalence_counit_iso Hg).
- exact (k · left_adjoint_right_adjoint Hg).
- simple refine (_ ,, _) ; cbn.
+ refine (rassociator _ _ _ • (k ◃ _) • runitor _).
exact (left_adjoint_counit Hg).
+ cbn.
use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
apply (left_equivalence_counit_iso Hg).
Definition trivial_lift_2cell
{w : B₁}
{z : B₂}
{h h' : w --> x₁}
{k k' : z --> y₂}
(γ : h ==> h')
(δ : k ==> k')
(Lh : lift_1cell_factor (trivial_displayed_bicat B₁ B₂) g k)
(Lh' : lift_1cell_factor (trivial_displayed_bicat B₁ B₂) g k')
: lift_2cell_factor (trivial_displayed_bicat B₁ B₂) g (δ := γ) δ Lh Lh'.
Show proof.
use iscontraprop1.
- abstract
(use invproofirrelevance ;
intros φ₁ φ₂ ;
induction φ₁ as [ φ₁ ψ₁ ] ;
induction φ₂ as [ φ₂ ψ₂ ] ;
use subtypePath ; [ intro ; apply cellset_property | ] ;
cbn in * ;
rewrite transportf_const in ψ₁, ψ₂ ;
cbn in * ;
apply (fully_faithful_1cell_faithful (adj_equiv_fully_faithful Hg)) ;
pose (p₂ := trivial_disp_invertible_to_invertible_2cell (pr2 Lh')) ;
use (vcomp_rcancel p₂) ; [ apply p₂ | ] ;
exact (ψ₁ @ !ψ₂)).
- simple refine (_ ,, _) ; cbn.
+ exact (fully_faithful_1cell_inv_map
(adj_equiv_fully_faithful Hg)
(pr12 Lh
• δ
• trivial_disp_invertible_to_invertible_2cell (pr2 Lh')^-1)).
+ abstract
(rewrite transportf_const ; cbn ;
etrans ;
[ apply maponpaths_2 ;
apply (fully_faithful_1cell_inv_map_eq (adj_equiv_fully_faithful Hg))
| ] ;
cbn in * ;
rewrite !vassocl ;
apply maponpaths ;
refine (_ @ id2_right _) ;
apply maponpaths ;
apply (vcomp_linv
(trivial_disp_invertible_to_invertible_2cell (pr2 Lh')))).
- abstract
(use invproofirrelevance ;
intros φ₁ φ₂ ;
induction φ₁ as [ φ₁ ψ₁ ] ;
induction φ₂ as [ φ₂ ψ₂ ] ;
use subtypePath ; [ intro ; apply cellset_property | ] ;
cbn in * ;
rewrite transportf_const in ψ₁, ψ₂ ;
cbn in * ;
apply (fully_faithful_1cell_faithful (adj_equiv_fully_faithful Hg)) ;
pose (p₂ := trivial_disp_invertible_to_invertible_2cell (pr2 Lh')) ;
use (vcomp_rcancel p₂) ; [ apply p₂ | ] ;
exact (ψ₁ @ !ψ₂)).
- simple refine (_ ,, _) ; cbn.
+ exact (fully_faithful_1cell_inv_map
(adj_equiv_fully_faithful Hg)
(pr12 Lh
• δ
• trivial_disp_invertible_to_invertible_2cell (pr2 Lh')^-1)).
+ abstract
(rewrite transportf_const ; cbn ;
etrans ;
[ apply maponpaths_2 ;
apply (fully_faithful_1cell_inv_map_eq (adj_equiv_fully_faithful Hg))
| ] ;
cbn in * ;
rewrite !vassocl ;
apply maponpaths ;
refine (_ @ id2_right _) ;
apply maponpaths ;
apply (vcomp_linv
(trivial_disp_invertible_to_invertible_2cell (pr2 Lh')))).
Definition trivial_cartesian_1cell
: cartesian_1cell (trivial_displayed_bicat B₁ B₂) g.
Show proof.
End TrivialCartesianOneCell.
Section Cartesian1CellToAdjEquiv.
Context {x₁ x₂ : B₁}
{y₁ : trivial_displayed_bicat B₁ B₂ x₁}
{y₂ : trivial_displayed_bicat B₁ B₂ x₂}
(f : x₁ --> x₂)
(g : y₁ -->[ f ] y₂)
(Hg : cartesian_1cell (trivial_displayed_bicat B₁ B₂) g).
Let inv : y₂ --> y₁ := pr1 (pr1 Hg x₁ y₂ (id₁ _) (id₁ _)).
Let ε : inv · g ==> id₁ y₂ := pr12 (pr1 Hg x₁ y₂ (id₁ _) (id₁ _)).
Let εinv
: is_invertible_2cell ε
:= trivial_disp_invertible_to_invertible_2cell
(pr2 (pr1 Hg x₁ y₂ (id₁ _) (id₁ _))).
Local Definition unit_help_lift₁
: lift_1cell_factor (trivial_displayed_bicat B₁ B₂) (h := id₁ _) g g.
Show proof.
simple refine (_ ,, _ ,, _) ; cbn.
- apply id₁.
- apply lunitor.
- use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
- apply id₁.
- apply lunitor.
- use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
Local Definition unit_help_lift₂
: lift_1cell_factor (trivial_displayed_bicat B₁ B₂) (h := id₁ _) g g.
Show proof.
simple refine (_ ,, _ ,, _) ; cbn.
- exact (g · inv).
- exact (rassociator _ _ _ • (g ◃ ε) • runitor _).
- use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
- exact (g · inv).
- exact (rassociator _ _ _ • (g ◃ ε) • runitor _).
- use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
Local Definition unit
: id₁ y₁ ==> g · inv
:= cartesian_1cell_lift_2cell
_ _
Hg
(δ := id2 _) (id2 _)
unit_help_lift₁ unit_help_lift₂.
Definition trivial_cartesian_1cell_left_adj_data
: left_adjoint_data g
:= (inv ,, unit ,, ε).
Definition trivial_cartesian_1cell_left_equiv_axioms
: left_equivalence_axioms trivial_cartesian_1cell_left_adj_data.
Show proof.
split.
- use (trivial_is_disp_invertible_to_is_invertible_2cell (α := id2 (id₁ _))).
+ is_iso.
+ refine (cartesian_1cell_lift_2cell_invertible
_ Hg _ _
unit_help_lift₁
unit_help_lift₂).
use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
- exact εinv.
- use (trivial_is_disp_invertible_to_is_invertible_2cell (α := id2 (id₁ _))).
+ is_iso.
+ refine (cartesian_1cell_lift_2cell_invertible
_ Hg _ _
unit_help_lift₁
unit_help_lift₂).
use trivial_is_invertible_2cell_to_is_disp_invertible.
is_iso.
- exact εinv.
Definition trivial_cartesian_1cell_is_adj_equiv
: left_adjoint_equivalence g.
Show proof.
use equiv_to_adjequiv.
simple refine (_ ,, _).
- exact trivial_cartesian_1cell_left_adj_data.
- exact trivial_cartesian_1cell_left_equiv_axioms.
End Cartesian1CellToAdjEquiv.simple refine (_ ,, _).
- exact trivial_cartesian_1cell_left_adj_data.
- exact trivial_cartesian_1cell_left_equiv_axioms.
Definition trivial_global_cleaving
: global_cleaving (trivial_displayed_bicat B₁ B₂).
Show proof.
intros x₁ x₂ y₁ y₂.
simple refine (y₁ ,, id₁ _ ,, _) ; cbn.
apply trivial_cartesian_1cell.
apply (internal_adjoint_equivalence_identity y₁).
simple refine (y₁ ,, id₁ _ ,, _) ; cbn.
apply trivial_cartesian_1cell.
apply (internal_adjoint_equivalence_identity y₁).
Definition trivial_lwhisker_cartesian
: lwhisker_cartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
intros ? ? ? ? ? ? ? ? ? ? ? ? ? ? Hαα.
apply trivial_invertible_is_cartesian_2cell.
cbn.
is_iso.
apply trivial_cartesian_2cell_is_invertible.
apply Hαα.
apply trivial_invertible_is_cartesian_2cell.
cbn.
is_iso.
apply trivial_cartesian_2cell_is_invertible.
apply Hαα.
Definition trivial_rwhisker_cartesian
: rwhisker_cartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
intros ? ? ? ? ? ? ? ? ? ? ? ? ? ? Hαα.
apply trivial_invertible_is_cartesian_2cell.
cbn.
is_iso.
apply trivial_cartesian_2cell_is_invertible.
apply Hαα.
apply trivial_invertible_is_cartesian_2cell.
cbn.
is_iso.
apply trivial_cartesian_2cell_is_invertible.
apply Hαα.
Definition trivial_lwhisker_opcartesian
: lwhisker_opcartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
intros ? ? ? ? ? ? ? ? ? ? ? ? ? ? Hαα.
apply trivial_invertible_is_opcartesian_2cell.
cbn.
is_iso.
apply trivial_opcartesian_2cell_is_invertible.
apply Hαα.
apply trivial_invertible_is_opcartesian_2cell.
cbn.
is_iso.
apply trivial_opcartesian_2cell_is_invertible.
apply Hαα.
Definition trivial_rwhisker_opcartesian
: rwhisker_opcartesian (trivial_displayed_bicat B₁ B₂).
Show proof.
intros ? ? ? ? ? ? ? ? ? ? ? ? ? ? Hαα.
apply trivial_invertible_is_opcartesian_2cell.
cbn.
is_iso.
apply trivial_opcartesian_2cell_is_invertible.
apply Hαα.
apply trivial_invertible_is_opcartesian_2cell.
cbn.
is_iso.
apply trivial_opcartesian_2cell_is_invertible.
apply Hαα.
Definition trivial_cleaving_of_bicats
: cleaving_of_bicats (trivial_displayed_bicat B₁ B₂).
Show proof.
repeat split.
- exact trivial_local_cleaving.
- exact trivial_global_cleaving.
- exact trivial_lwhisker_cartesian.
- exact trivial_rwhisker_cartesian.
End ConstantCleaving.- exact trivial_local_cleaving.
- exact trivial_global_cleaving.
- exact trivial_lwhisker_cartesian.
- exact trivial_rwhisker_cartesian.