Library UniMath.CategoryTheory.Bicategories.Bicategories.Examples.OpMorBicat
op1 bicategory.
Bicategory obtained by formally reversing 1-arrows.
Benedikt Ahrens, Marco Maggesi
June 2018
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Categories.
Require Import UniMath.CategoryTheory.functor_categories.
Require Import UniMath.CategoryTheory.opp_precat.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Bicat. Import Bicat.Notations.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Invertible_2cells.
Require Import UniMath.CategoryTheory.Bicategories.Bicategories.Adjunctions.
Local Open Scope cat.
Definition op1_2cell_struct (C : prebicat_1_id_comp_cells)
: prebicat_2cell_struct (opp_precat_data C)
:= λ (a b : C) (f g : C ⟦ b, a ⟧), f ==> g.
Definition op1_prebicat_1_id_comp_cells (C : prebicat_1_id_comp_cells)
: prebicat_1_id_comp_cells
:= (opp_precat_data C),, op1_2cell_struct C.
Definition op1_prebicat_data (C : prebicat_data)
: prebicat_data.
Proof.
∃ (op1_prebicat_1_id_comp_cells C).
red. cbn. unfold op1_2cell_struct.
repeat use dirprodpair; intros until 0.
- apply id2.
- apply runitor.
- apply lunitor.
- apply rinvunitor.
- apply linvunitor.
- apply lassociator.
- apply rassociator.
- intros α β. exact ( α • β ).
- intros α. exact (α ▹ f).
- intros α. exact (g ◃ α).
Defined.
Definition op1_prebicat_laws (C : prebicat) : prebicat_laws (op1_prebicat_data C).
Proof.
red. cbn. unfold op1_2cell_struct. cbn.
repeat use tpair; cbn; intros until 0.
- apply id2_left.
- apply id2_right.
- apply vassocr.
- apply id2_rwhisker.
- apply lwhisker_id2.
- apply rwhisker_vcomp.
- apply lwhisker_vcomp.
- apply vcomp_runitor.
- apply vcomp_lunitor.
- apply rwhisker_rwhisker_alt.
- apply pathsinv0, rwhisker_lwhisker_rassociator.
- apply lwhisker_lwhisker_rassociator.
- apply pathsinv0, vcomp_whisker.
- apply runitor_rinvunitor.
- apply rinvunitor_runitor.
- apply lunitor_linvunitor.
- apply linvunitor_lunitor.
- apply rassociator_lassociator.
- apply lassociator_rassociator.
- apply lunitor_lwhisker.
- apply rassociator_rassociator.
Qed.
Definition op1_prebicat (C : prebicat) : prebicat
:= op1_prebicat_data C,, op1_prebicat_laws C.
Definition op1_isaset_cells (C : bicat) : isaset_cells (op1_prebicat C)
:= λ (a b : C) (f g : C ⟦ b, a ⟧), cellset_property f g.
Definition op1_bicat (C : bicat) : bicat
:= op1_prebicat C,, op1_isaset_cells C.
Definition op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell
{C : bicat}
{X Y : op1_bicat C}
{f g : X --> Y}
(α : f ==> g)
: is_invertible_2cell α → @is_invertible_2cell C Y X f g α.
Proof.
intros Hα.
use tpair.
- apply Hα.
- split ; apply Hα.
Defined.
Definition bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell
{C : bicat}
{X Y : op1_bicat C}
{f g : X --> Y}
(α : f ==> g)
: @is_invertible_2cell C Y X f g α → is_invertible_2cell α.
Proof.
intros Hα.
use tpair.
- apply Hα.
- split ; apply Hα.
Defined.
Definition op1_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell
{C : bicat}
{X Y : op1_bicat C}
{f g : X --> Y}
(α : f ==> g)
: @is_invertible_2cell C Y X f g α ≃ is_invertible_2cell α.
Proof.
use weqimplimpl.
- exact (bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell α).
- exact (op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell α).
- apply isaprop_is_invertible_2cell.
- apply isaprop_is_invertible_2cell.
Defined.
Definition bicat_invertible_2cell_is_op1_bicat_invertible_2cell
{C : bicat}
{X Y : op1_bicat C}
(f g : X --> Y)
: @invertible_2cell C Y X f g ≃ invertible_2cell f g.
Proof.
use weqfibtototal.
intro α.
apply op1_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
Defined.
Definition op1_bicat_left_adjoint_equivalence_to_bicat_left_adjoint_equivalence
{C : bicat}
{X Y : op1_bicat C}
(f : X --> Y)
: left_adjoint_equivalence f → @left_adjoint_equivalence C Y X f.
Proof.
intros Hf.
use tpair.
- use tpair.
+ exact (left_adjoint_right_adjoint Hf).
+ split.
× exact ((left_equivalence_counit_iso Hf)^-1).
× exact ((left_equivalence_unit_iso Hf)^-1).
- split ; split.
+ use inv_cell_eq ; cbn.
× is_iso.
** apply op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
** apply op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
× is_iso.
× cbn.
rewrite !vassocr.
exact (internal_triangle1 Hf).
+ use inv_cell_eq ; cbn.
× is_iso.
** apply op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
** apply op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
× is_iso.
× cbn.
rewrite !vassocr.
exact (internal_triangle2 Hf).
+ cbn.
apply op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
+ cbn.
apply op1_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
Defined.
Definition bicat_left_adjoint_equivalence_to_op1_bicat_left_adjoint_equivalence
{C : bicat}
{X Y : op1_bicat C}
(f : X --> Y)
: @left_adjoint_equivalence C Y X f → left_adjoint_equivalence f.
Proof.
intros Hf.
use tpair.
- use tpair.
+ exact (left_adjoint_right_adjoint Hf).
+ split.
× exact ((left_equivalence_counit_iso Hf)^-1).
× exact ((left_equivalence_unit_iso Hf)^-1).
- split ; split.
+ use inv_cell_eq ; cbn.
× apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell.
is_iso.
× apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell.
is_iso.
× cbn.
rewrite !vassocr.
exact (internal_triangle1 Hf).
+ use inv_cell_eq ; cbn.
× apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell.
is_iso.
× apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell.
is_iso.
× cbn.
rewrite !vassocr.
exact (internal_triangle2 Hf).
+ cbn.
apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell.
is_iso.
+ cbn.
apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell.
is_iso.
Defined.
Definition op1_bicat_left_adjoint_equivalence_is_bicat_left_adjoint_equivalence
{C : bicat}
{X Y : op1_bicat C}
(f : X --> Y)
: @left_adjoint_equivalence C Y X f ≃ left_adjoint_equivalence f.
Proof.
use weqpair.
- exact (bicat_left_adjoint_equivalence_to_op1_bicat_left_adjoint_equivalence f).
- use isweq_iso.
+ exact (op1_bicat_left_adjoint_equivalence_to_bicat_left_adjoint_equivalence f).
+ intros x.
use subtypeEquality.
× intro.
do 2 apply isapropdirprod ; try (apply C) ; apply isaprop_is_invertible_2cell.
× reflexivity.
+ intros x.
use subtypeEquality.
× intro.
do 2 apply isapropdirprod ; try (apply C) ; apply isaprop_is_invertible_2cell.
× reflexivity.
Defined.
Definition bicat_adjoint_equivalence_is_op1_bicat_adjoint_equivalence
{C : bicat}
(X Y : op1_bicat C)
: @adjoint_equivalence C Y X ≃ adjoint_equivalence X Y.
Proof.
use weqfibtototal.
intro α.
apply op1_bicat_left_adjoint_equivalence_is_bicat_left_adjoint_equivalence.
Defined.