Library UniMath.Bicategories.Core.Examples.OpCellBicat
op2 bicategory.
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.opp_precat.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Adjunctions.
Local Open Scope cat.
Definition op2_2cell_struct (C : prebicat_1_id_comp_cells)
: prebicat_2cell_struct C
:= λ (a b : C) (f g : C ⟦ a, b ⟧), g ==> f.
Definition op2_prebicat_1_id_comp_cells (C : prebicat_1_id_comp_cells)
: prebicat_1_id_comp_cells
:= (C:precategory_data),, op2_2cell_struct C.
Definition op2_prebicat_data (C : prebicat_data)
: prebicat_data.
Proof.
∃ (op2_prebicat_1_id_comp_cells C).
repeat use make_dirprod.
- intros; apply id2.
- intros; cbn. apply linvunitor.
- intros; cbn. apply rinvunitor.
- intros; cbn. apply lunitor.
- intros; cbn. apply runitor.
- intros; cbn. apply lassociator.
- intros; cbn. apply rassociator.
- intros; cbn. apply ( X0 • X ).
- intros; cbn. apply ( f ◃ X ).
- cbn; intros. apply (X ▹ g).
Defined.
Section op2.
Variable C : prebicat.
Definition op2_prebicat_laws : prebicat_laws (op2_prebicat_data C).
Proof.
repeat split; intros; cbn.
- apply id2_right.
- apply id2_left.
- apply (!vassocr _ _ _ ).
- apply lwhisker_id2.
- apply id2_rwhisker.
- apply lwhisker_vcomp.
- apply rwhisker_vcomp.
- use lhs_left_invert_cell; [ apply is_invertible_2cell_linvunitor |].
cbn.
apply pathsinv0.
etrans. apply vassocr.
use lhs_right_invert_cell; [ apply is_invertible_2cell_linvunitor |].
cbn.
apply pathsinv0. apply vcomp_lunitor.
- use lhs_left_invert_cell; [ apply is_invertible_2cell_rinvunitor |].
cbn.
apply pathsinv0.
etrans. apply vassocr.
use lhs_right_invert_cell; [ apply is_invertible_2cell_rinvunitor |].
cbn.
apply pathsinv0. apply vcomp_runitor.
- apply lassociator_to_rassociator_pre.
apply pathsinv0.
etrans. apply (vassocr _ _ _ ).
apply lassociator_to_rassociator_post.
apply pathsinv0. apply lwhisker_lwhisker.
- apply lassociator_to_rassociator_pre.
apply pathsinv0.
etrans. apply (vassocr _ _ _ ).
apply lassociator_to_rassociator_post.
apply pathsinv0. apply rwhisker_lwhisker.
- apply pathsinv0, lassociator_to_rassociator_pre.
apply pathsinv0.
etrans. apply (vassocr _ _ _ ).
apply lassociator_to_rassociator_post.
apply rwhisker_rwhisker.
- apply (!vcomp_whisker _ _ ).
- apply lunitor_linvunitor.
- apply linvunitor_lunitor.
- apply runitor_rinvunitor.
- apply rinvunitor_runitor.
- apply lassociator_rassociator.
- apply rassociator_lassociator.
- use lhs_left_invert_cell.
{ use is_invertible_2cell_rwhisker.
apply is_invertible_2cell_rinvunitor.
} cbn.
apply pathsinv0.
use lhs_right_invert_cell.
{ use is_invertible_2cell_lwhisker.
apply is_invertible_2cell_linvunitor.
} cbn.
apply pathsinv0.
apply lassociator_to_rassociator_pre.
apply pathsinv0, runitor_rwhisker.
- use lhs_left_invert_cell.
{ use is_invertible_2cell_rwhisker.
apply is_invertible_2cell_rassociator.
} cbn.
apply pathsinv0.
etrans. apply vassocr.
use lhs_right_invert_cell.
{ apply is_invertible_2cell_rassociator.
} cbn.
use lhs_right_invert_cell.
{ apply is_invertible_2cell_rassociator.
} cbn.
apply pathsinv0.
repeat rewrite <- vassocr.
use lhs_left_invert_cell.
{ apply is_invertible_2cell_rassociator.
} cbn.
use lhs_left_invert_cell.
{ apply is_invertible_2cell_lwhisker.
apply is_invertible_2cell_rassociator.
} cbn.
apply pathsinv0.
rewrite vassocr.
apply lassociator_lassociator.
Qed.
Definition op2_prebicat : prebicat := op2_prebicat_data C ,, op2_prebicat_laws.
End op2.
Definition op2_isaset_cells (C : bicat) : isaset_cells (op2_prebicat C)
:= λ (a b : C) (f g : C ⟦ a, b ⟧), cellset_property g f.
Definition op2_bicat (C : bicat) : bicat
:= op2_prebicat C,, op2_isaset_cells C.
Definition op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell
{C : bicat}
{X Y : op2_bicat C}
{f g : X --> Y}
(α : f ==> g)
: is_invertible_2cell α → @is_invertible_2cell C X Y g f α.
Proof.
intros Hα.
use tpair.
- apply Hα.
- split ; apply Hα.
Defined.
Definition bicat_is_invertible_2cell_to_op2_bicat_is_invertible_2cell
{C : bicat}
{X Y : op2_bicat C}
{f g : X --> Y}
(α : f ==> g)
: @is_invertible_2cell C X Y g f α → is_invertible_2cell α.
Proof.
intros Hα.
use tpair.
- apply Hα.
- split ; apply Hα.
Defined.
Definition op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell
{C : bicat}
{X Y : op2_bicat C}
{f g : X --> Y}
(α : f ==> g)
: @is_invertible_2cell C X Y g f α ≃ is_invertible_2cell α.
Proof.
use weqimplimpl.
- exact (bicat_is_invertible_2cell_to_op2_bicat_is_invertible_2cell α).
- exact (op2_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_op2_bicat_invertible_2cell
{C : bicat}
{X Y : op2_bicat C}
(f g : X --> Y)
: @invertible_2cell C X Y g f ≃ invertible_2cell f g.
Proof.
use weqfibtototal.
intro α.
apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
Defined.
Definition op2_bicat_left_adjoint_equivalence_to_bicat_left_adjoint_equivalence
{C : bicat}
{X Y : op2_bicat C}
(f : X --> Y)
: left_adjoint_equivalence f → @left_adjoint_equivalence C X Y f.
Proof.
intros Hf.
use tpair.
- use tpair.
+ exact (left_adjoint_right_adjoint Hf).
+ split.
× exact ((left_equivalence_unit_iso Hf)^-1).
× exact ((left_equivalence_counit_iso Hf)^-1).
- split ; split.
+ use inv_cell_eq ; cbn.
× is_iso.
** apply op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
** apply op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
× is_iso.
× exact (internal_triangle1 Hf).
+ use inv_cell_eq ; cbn.
× is_iso.
** apply op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
** apply op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
× is_iso.
× exact (internal_triangle2 Hf).
+ cbn.
apply op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
+ cbn.
apply op2_bicat_is_invertible_2cell_to_bicat_is_invertible_2cell.
is_iso.
Defined.
Definition bicat_left_adjoint_equivalence_to_op2_bicat_left_adjoint_equivalence
{C : bicat}
{X Y : op2_bicat C}
(f : X --> Y)
: @left_adjoint_equivalence C X Y f → left_adjoint_equivalence f.
Proof.
intros Hf.
use tpair.
- use tpair.
+ exact (left_adjoint_right_adjoint Hf).
+ split.
× exact ((left_equivalence_unit_iso Hf)^-1).
× exact ((left_equivalence_counit_iso Hf)^-1).
- split ; split.
+ use inv_cell_eq ; cbn.
× apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
is_iso.
× apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
is_iso.
× exact (internal_triangle1 Hf).
+ use inv_cell_eq ; cbn.
× apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
is_iso.
× apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
is_iso.
× exact (internal_triangle2 Hf).
+ cbn.
apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
is_iso.
+ cbn.
apply op2_bicat_is_invertible_2cell_is_bicat_is_invertible_2cell.
is_iso.
Defined.
Definition op2_bicat_left_adjoint_equivalence_is_bicat_left_adjoint_equivalence
{C : bicat}
{X Y : op2_bicat C}
(f : X --> Y)
: @left_adjoint_equivalence C X Y f ≃ left_adjoint_equivalence f.
Proof.
use make_weq.
- exact (bicat_left_adjoint_equivalence_to_op2_bicat_left_adjoint_equivalence f).
- use isweq_iso.
+ exact (op2_bicat_left_adjoint_equivalence_to_bicat_left_adjoint_equivalence f).
+ intros x.
use subtypePath.
× intro.
do 2 apply isapropdirprod ; try (apply C) ; apply isaprop_is_invertible_2cell.
× reflexivity.
+ intros x.
use subtypePath.
× intro.
do 2 apply isapropdirprod ; try (apply C) ; apply isaprop_is_invertible_2cell.
× reflexivity.
Defined.
Definition bicat_adjoint_equivalence_is_op2_bicat_adjoint_equivalence
{C : bicat}
(X Y : op2_bicat C)
: @adjoint_equivalence C X Y ≃ adjoint_equivalence X Y.
Proof.
use weqfibtototal.
intro α.
apply op2_bicat_left_adjoint_equivalence_is_bicat_left_adjoint_equivalence.
Defined.