Library UniMath.Bicategories.DisplayedBicats.FiberBicategory.DisplayMapFiber
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Logic.DisplayMapBicat.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayMapBicatToDispBicat.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayMapBicatSlice.
Require Import UniMath.Bicategories.DisplayedBicats.FiberBicategory.
Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.Transformations.PseudoTransformation.
Require Import UniMath.Bicategories.Modifications.Modification.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.
Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence.
Require Import UniMath.Bicategories.DisplayedBicats.ExamplesOfCleavings.DisplayMapBicatCleaving.
Local Open Scope cat.
Local Open Scope bicategory_scope.
Section FiberOfDisplayMap.
Context {B : bicat}
(D : arrow_subbicat B)
(b : B).
Let DD : disp_bicat B := disp_map_bicat_to_disp_bicat D.
Let slice : bicat := disp_map_slice_bicat D b.
Let fiber : bicat
:= strict_fiber_bicat DD (arrow_subbicat_local_iso_cleaving D) b.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Logic.DisplayMapBicat.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayMapBicatToDispBicat.
Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayMapBicatSlice.
Require Import UniMath.Bicategories.DisplayedBicats.FiberBicategory.
Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor.
Import PseudoFunctor.Notations.
Require Import UniMath.Bicategories.Transformations.PseudoTransformation.
Require Import UniMath.Bicategories.Modifications.Modification.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity.
Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.
Require Import UniMath.Bicategories.PseudoFunctors.Biequivalence.
Require Import UniMath.Bicategories.DisplayedBicats.ExamplesOfCleavings.DisplayMapBicatCleaving.
Local Open Scope cat.
Local Open Scope bicategory_scope.
Section FiberOfDisplayMap.
Context {B : bicat}
(D : arrow_subbicat B)
(b : B).
Let DD : disp_bicat B := disp_map_bicat_to_disp_bicat D.
Let slice : bicat := disp_map_slice_bicat D b.
Let fiber : bicat
:= strict_fiber_bicat DD (arrow_subbicat_local_iso_cleaving D) b.
1. Calculation of operations in fiber
Definition comp_disp_map_fiber
{x y : fiber}
{f g h : x --> y}
(α : f ==> g)
(β : g ==> h)
: pr1 (α • β) = pr1 α • pr1 β.
Show proof.
Definition lunitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (lunitor f) = lunitor (pr1 f).
Show proof.
Definition linvunitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (linvunitor f) = linvunitor (pr1 f).
Show proof.
Definition runitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (runitor f) = runitor (pr1 f).
Show proof.
Definition rinvunitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (rinvunitor f) = rinvunitor (pr1 f).
Show proof.
Definition lassociator_disp_map_fiber
{w x y z : fiber}
(f : w --> x)
(g : x --> y)
(h : y --> z)
: pr1 (lassociator f g h) = lassociator (pr1 f) (pr1 g) (pr1 h).
Show proof.
Definition rassociator_disp_map_fiber
{w x y z : fiber}
(f : w --> x)
(g : x --> y)
(h : y --> z)
: pr1 (rassociator f g h) = rassociator (pr1 f) (pr1 g) (pr1 h).
Show proof.
Definition lwhisker_disp_map_fiber
{x y z : fiber}
(f : x --> y)
{g h : y --> z}
(α : g ==> h)
: pr1 (f ◃ α) = _ ◃ pr1 α.
Show proof.
Definition rwhisker_disp_map_fiber
{x y z : fiber}
{f g : x --> y}
(α : f ==> g)
(h : y --> z)
: pr1 (α ▹ h) = pr1 α ▹ _.
Show proof.
Definition path_2cell_disp_map_fiber
{x y : fiber}
{f g : x --> y}
(α β : f ==> g)
(p : pr1 α = pr1 β)
: α = β.
Show proof.
{x y : fiber}
{f g h : x --> y}
(α : f ==> g)
(β : g ==> h)
: pr1 (α • β) = pr1 α • pr1 β.
Show proof.
Definition lunitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (lunitor f) = lunitor (pr1 f).
Show proof.
Definition linvunitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (linvunitor f) = linvunitor (pr1 f).
Show proof.
Definition runitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (runitor f) = runitor (pr1 f).
Show proof.
Definition rinvunitor_disp_map_fiber
{x y : fiber}
(f : x --> y)
: pr1 (rinvunitor f) = rinvunitor (pr1 f).
Show proof.
Definition lassociator_disp_map_fiber
{w x y z : fiber}
(f : w --> x)
(g : x --> y)
(h : y --> z)
: pr1 (lassociator f g h) = lassociator (pr1 f) (pr1 g) (pr1 h).
Show proof.
etrans.
{
apply transportf_disp_map_bicat_cell.
}
cbn.
rewrite lwhisker_id2, id2_rwhisker, !id2_left, !id2_right.
apply idpath.
{
apply transportf_disp_map_bicat_cell.
}
cbn.
rewrite lwhisker_id2, id2_rwhisker, !id2_left, !id2_right.
apply idpath.
Definition rassociator_disp_map_fiber
{w x y z : fiber}
(f : w --> x)
(g : x --> y)
(h : y --> z)
: pr1 (rassociator f g h) = rassociator (pr1 f) (pr1 g) (pr1 h).
Show proof.
etrans.
{
apply transportf_disp_map_bicat_cell.
}
cbn.
rewrite lwhisker_id2, id2_rwhisker, !id2_left, !id2_right.
apply idpath.
{
apply transportf_disp_map_bicat_cell.
}
cbn.
rewrite lwhisker_id2, id2_rwhisker, !id2_left, !id2_right.
apply idpath.
Definition lwhisker_disp_map_fiber
{x y z : fiber}
(f : x --> y)
{g h : y --> z}
(α : g ==> h)
: pr1 (f ◃ α) = _ ◃ pr1 α.
Show proof.
Definition rwhisker_disp_map_fiber
{x y z : fiber}
{f g : x --> y}
(α : f ==> g)
(h : y --> z)
: pr1 (α ▹ h) = pr1 α ▹ _.
Show proof.
Definition path_2cell_disp_map_fiber
{x y : fiber}
{f g : x --> y}
(α β : f ==> g)
(p : pr1 α = pr1 β)
: α = β.
Show proof.
2. To the fiber
Definition to_fiber_disp_map_data
: psfunctor_data slice fiber.
Show proof.
Definition to_fiber_disp_map_laws
: psfunctor_laws to_fiber_disp_map_data.
Show proof.
Definition to_fiber_disp_map
: psfunctor slice fiber.
Show proof.
: psfunctor_data slice fiber.
Show proof.
use make_psfunctor_data.
- exact (λ f, f).
- exact (λ a₁ a₂ g,
pr1 g
,,
pr12 g
,,
comp_of_invertible_2cell
(runitor_invertible_2cell _)
(pr22 g)).
- refine (λ a₁ a₂ g₁ g₂ β, pr1 β ,, _).
abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite !vassocl ;
apply maponpaths ;
exact (pr2 β)).
- refine (λ a, id2 _ ,, _).
abstract
(cbn ;
rewrite id2_rwhisker, id2_right ;
rewrite lwhisker_id2, id2_left ;
apply idpath).
- refine (λ a₁ a₂ a₃ g₁ g₂, id2 _ ,, _).
abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite id2_rwhisker, id2_right ;
rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp ;
rewrite !vassocr ;
do 2 apply maponpaths_2 ;
rewrite !vassocl ;
rewrite <- lunitor_lwhisker ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite lassociator_rassociator ;
rewrite id2_left ;
rewrite !vassocr ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
rewrite !vassocl ;
rewrite runitor_triangle ;
rewrite vcomp_runitor ;
apply idpath).
- exact (λ f, f).
- exact (λ a₁ a₂ g,
pr1 g
,,
pr12 g
,,
comp_of_invertible_2cell
(runitor_invertible_2cell _)
(pr22 g)).
- refine (λ a₁ a₂ g₁ g₂ β, pr1 β ,, _).
abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite !vassocl ;
apply maponpaths ;
exact (pr2 β)).
- refine (λ a, id2 _ ,, _).
abstract
(cbn ;
rewrite id2_rwhisker, id2_right ;
rewrite lwhisker_id2, id2_left ;
apply idpath).
- refine (λ a₁ a₂ a₃ g₁ g₂, id2 _ ,, _).
abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite id2_rwhisker, id2_right ;
rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp ;
rewrite !vassocr ;
do 2 apply maponpaths_2 ;
rewrite !vassocl ;
rewrite <- lunitor_lwhisker ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite lassociator_rassociator ;
rewrite id2_left ;
rewrite !vassocr ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
rewrite !vassocl ;
rewrite runitor_triangle ;
rewrite vcomp_runitor ;
apply idpath).
Definition to_fiber_disp_map_laws
: psfunctor_laws to_fiber_disp_map_data.
Show proof.
refine (_ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _) ;
intro ; intros ;
use path_2cell_disp_map_fiber.
- apply idpath.
- rewrite comp_disp_map_fiber.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite lunitor_disp_map_fiber.
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite id2_rwhisker, !id2_left.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite runitor_disp_map_fiber.
rewrite lwhisker_disp_map_fiber.
cbn.
rewrite lwhisker_id2, !id2_left.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite lwhisker_disp_map_fiber, rwhisker_disp_map_fiber.
rewrite lassociator_disp_map_fiber.
cbn.
rewrite lwhisker_id2, id2_rwhisker.
rewrite !id2_left, !id2_right.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite lwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
intro ; intros ;
use path_2cell_disp_map_fiber.
- apply idpath.
- rewrite comp_disp_map_fiber.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite lunitor_disp_map_fiber.
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite id2_rwhisker, !id2_left.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite runitor_disp_map_fiber.
rewrite lwhisker_disp_map_fiber.
cbn.
rewrite lwhisker_id2, !id2_left.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite lwhisker_disp_map_fiber, rwhisker_disp_map_fiber.
rewrite lassociator_disp_map_fiber.
cbn.
rewrite lwhisker_id2, id2_rwhisker.
rewrite !id2_left, !id2_right.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite lwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
- rewrite !comp_disp_map_fiber.
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
Definition to_fiber_disp_map
: psfunctor slice fiber.
Show proof.
use make_psfunctor.
- exact to_fiber_disp_map_data.
- exact to_fiber_disp_map_laws.
- split.
+ intros.
use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible ; cbn.
is_iso.
+ intros.
use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible ; cbn.
is_iso.
- exact to_fiber_disp_map_data.
- exact to_fiber_disp_map_laws.
- split.
+ intros.
use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible ; cbn.
is_iso.
+ intros.
use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible ; cbn.
is_iso.
3. From the fiber
Definition from_fiber_disp_map_data
: psfunctor_data fiber slice.
Show proof.
Definition from_fiber_disp_map_laws
: psfunctor_laws from_fiber_disp_map_data.
Show proof.
Definition from_fiber_disp_map
: psfunctor fiber slice.
Show proof.
: psfunctor_data fiber slice.
Show proof.
use make_psfunctor_data.
- exact (λ f, f).
- exact (λ a₁ a₂ g,
pr1 g
,,
pr12 g
,,
comp_of_invertible_2cell
(rinvunitor_invertible_2cell _)
(pr22 g)).
- refine (λ a₁ a₂ g₁ g₂ β, pr1 β ,, _).
abstract
(cbn ;
rewrite !vassocl ;
refine (maponpaths (λ z, _ • z) (pr2 β) @ _) ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
apply idpath).
- refine (λ a, id2 _ ,, _).
abstract
(cbn ;
rewrite id2_rwhisker ;
rewrite id2_right ;
rewrite !vassocr ;
rewrite rinvunitor_runitor ;
rewrite id2_left ;
apply idpath).
- refine (λ a₁ a₂ a₃ g₁ g₂, id2 _ ,, _).
abstract
(cbn ;
rewrite id2_rwhisker ;
rewrite id2_right ;
rewrite !vassocl ;
apply maponpaths ;
rewrite <- lwhisker_vcomp ;
rewrite !vassocr ;
do 2 apply maponpaths_2 ;
rewrite !lwhisker_hcomp ;
rewrite triangle_r_inv ;
rewrite <- rwhisker_hcomp, <- lwhisker_hcomp ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite left_unit_inv_assoc ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite rinvunitor_natural ;
rewrite <- rwhisker_hcomp ;
apply maponpaths_2 ;
refine (_ @ id2_right _) ;
use vcomp_move_L_pM ; [ is_iso | ] ;
cbn ;
use vcomp_move_R_Mp ; [ is_iso | ] ;
cbn ;
rewrite id2_left ;
rewrite <- runitor_triangle ;
rewrite runitor_lunitor_identity ;
rewrite lunitor_lwhisker ;
apply idpath).
- exact (λ f, f).
- exact (λ a₁ a₂ g,
pr1 g
,,
pr12 g
,,
comp_of_invertible_2cell
(rinvunitor_invertible_2cell _)
(pr22 g)).
- refine (λ a₁ a₂ g₁ g₂ β, pr1 β ,, _).
abstract
(cbn ;
rewrite !vassocl ;
refine (maponpaths (λ z, _ • z) (pr2 β) @ _) ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
apply idpath).
- refine (λ a, id2 _ ,, _).
abstract
(cbn ;
rewrite id2_rwhisker ;
rewrite id2_right ;
rewrite !vassocr ;
rewrite rinvunitor_runitor ;
rewrite id2_left ;
apply idpath).
- refine (λ a₁ a₂ a₃ g₁ g₂, id2 _ ,, _).
abstract
(cbn ;
rewrite id2_rwhisker ;
rewrite id2_right ;
rewrite !vassocl ;
apply maponpaths ;
rewrite <- lwhisker_vcomp ;
rewrite !vassocr ;
do 2 apply maponpaths_2 ;
rewrite !lwhisker_hcomp ;
rewrite triangle_r_inv ;
rewrite <- rwhisker_hcomp, <- lwhisker_hcomp ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite left_unit_inv_assoc ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite rinvunitor_natural ;
rewrite <- rwhisker_hcomp ;
apply maponpaths_2 ;
refine (_ @ id2_right _) ;
use vcomp_move_L_pM ; [ is_iso | ] ;
cbn ;
use vcomp_move_R_Mp ; [ is_iso | ] ;
cbn ;
rewrite id2_left ;
rewrite <- runitor_triangle ;
rewrite runitor_lunitor_identity ;
rewrite lunitor_lwhisker ;
apply idpath).
Definition from_fiber_disp_map_laws
: psfunctor_laws from_fiber_disp_map_data.
Show proof.
repeat split ;
intro ; intros ;
use eq_2cell_disp_map_slice.
- apply idpath.
- cbn -[vcomp2].
rewrite comp_disp_map_fiber.
cbn.
apply idpath.
- cbn -[lunitor].
rewrite lunitor_disp_map_fiber.
cbn.
rewrite id2_rwhisker, !id2_left.
apply idpath.
- cbn -[runitor].
rewrite runitor_disp_map_fiber.
cbn.
rewrite lwhisker_id2, !id2_left.
apply idpath.
- cbn -[lassociator].
rewrite lassociator_disp_map_fiber.
cbn.
rewrite id2_rwhisker, lwhisker_id2, !id2_left, !id2_right.
apply idpath.
- cbn -[lwhisker].
rewrite lwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
- cbn -[rwhisker].
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
intro ; intros ;
use eq_2cell_disp_map_slice.
- apply idpath.
- cbn -[vcomp2].
rewrite comp_disp_map_fiber.
cbn.
apply idpath.
- cbn -[lunitor].
rewrite lunitor_disp_map_fiber.
cbn.
rewrite id2_rwhisker, !id2_left.
apply idpath.
- cbn -[runitor].
rewrite runitor_disp_map_fiber.
cbn.
rewrite lwhisker_id2, !id2_left.
apply idpath.
- cbn -[lassociator].
rewrite lassociator_disp_map_fiber.
cbn.
rewrite id2_rwhisker, lwhisker_id2, !id2_left, !id2_right.
apply idpath.
- cbn -[lwhisker].
rewrite lwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
- cbn -[rwhisker].
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite id2_left, id2_right.
apply idpath.
Definition from_fiber_disp_map
: psfunctor fiber slice.
Show proof.
use make_psfunctor.
- exact from_fiber_disp_map_data.
- exact from_fiber_disp_map_laws.
- split ; intros.
+ use is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
+ use is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
- exact from_fiber_disp_map_data.
- exact from_fiber_disp_map_laws.
- split ; intros.
+ use is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
+ use is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
4. The unit
Definition to_fiber_disp_map_unit_data
: pstrans_data
(id_psfunctor _)
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map).
Show proof.
Definition to_fiber_disp_map_unit_is_pstrans
: is_pstrans to_fiber_disp_map_unit_data.
Show proof.
Definition to_fiber_disp_map_unit
: pstrans
(id_psfunctor _)
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map).
Show proof.
Definition to_fiber_disp_map_unit_inv_data
: pstrans_data
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map)
(id_psfunctor _).
Show proof.
Definition to_fiber_disp_map_unit_inv_is_pstrans
: is_pstrans to_fiber_disp_map_unit_inv_data.
Show proof.
Definition to_fiber_disp_map_unit_inv
: pstrans
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map)
(id_psfunctor _).
Show proof.
: pstrans_data
(id_psfunctor _)
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map).
Show proof.
use make_pstrans_data.
- exact (λ f, id₁ _ ,, id_pred_mor D _ ,, linvunitor_invertible_2cell _).
- cbn.
refine (λ f₁ f₂ g, _).
use make_invertible_2cell.
+ cbn.
simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite !vassocr ;
rewrite rinvunitor_runitor ;
rewrite id2_left ;
rewrite lwhisker_hcomp ;
rewrite <- linvunitor_natural ;
rewrite !vassocl ;
apply maponpaths ;
rewrite linvunitor_assoc ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocr ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
rewrite !vassocl ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
apply idpath).
+ apply is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
- exact (λ f, id₁ _ ,, id_pred_mor D _ ,, linvunitor_invertible_2cell _).
- cbn.
refine (λ f₁ f₂ g, _).
use make_invertible_2cell.
+ cbn.
simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite !vassocr ;
rewrite rinvunitor_runitor ;
rewrite id2_left ;
rewrite lwhisker_hcomp ;
rewrite <- linvunitor_natural ;
rewrite !vassocl ;
apply maponpaths ;
rewrite linvunitor_assoc ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocr ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
rewrite !vassocl ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
apply idpath).
+ apply is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
Definition to_fiber_disp_map_unit_is_pstrans
: is_pstrans to_fiber_disp_map_unit_data.
Show proof.
repeat split.
- intros x y f g α.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
apply idpath.
- intros x.
use eq_2cell_disp_map_slice.
cbn.
rewrite id2_left.
rewrite lwhisker_id2, id2_rwhisker.
rewrite id2_left, id2_right.
rewrite lunitor_runitor_identity, runitor_rinvunitor.
rewrite runitor_lunitor_identity, lunitor_linvunitor.
apply idpath.
- intros x y z f g.
use eq_2cell_disp_map_slice.
cbn.
rewrite id2_left.
rewrite lwhisker_id2, id2_rwhisker.
rewrite id2_left, id2_right.
rewrite !vassocl.
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn.
rewrite <- runitor_triangle.
rewrite !vassocl.
rewrite (maponpaths (λ z, _ • (_ • (_ • (_ • z)))) (vassocr _ _ _)).
rewrite lassociator_rassociator.
rewrite id2_left.
rewrite lwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite lunitor_triangle.
apply idpath.
- intros x y f g α.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
apply idpath.
- intros x.
use eq_2cell_disp_map_slice.
cbn.
rewrite id2_left.
rewrite lwhisker_id2, id2_rwhisker.
rewrite id2_left, id2_right.
rewrite lunitor_runitor_identity, runitor_rinvunitor.
rewrite runitor_lunitor_identity, lunitor_linvunitor.
apply idpath.
- intros x y z f g.
use eq_2cell_disp_map_slice.
cbn.
rewrite id2_left.
rewrite lwhisker_id2, id2_rwhisker.
rewrite id2_left, id2_right.
rewrite !vassocl.
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn.
rewrite <- runitor_triangle.
rewrite !vassocl.
rewrite (maponpaths (λ z, _ • (_ • (_ • (_ • z)))) (vassocr _ _ _)).
rewrite lassociator_rassociator.
rewrite id2_left.
rewrite lwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite lunitor_triangle.
apply idpath.
Definition to_fiber_disp_map_unit
: pstrans
(id_psfunctor _)
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map).
Show proof.
Definition to_fiber_disp_map_unit_inv_data
: pstrans_data
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map)
(id_psfunctor _).
Show proof.
use make_pstrans_data.
- exact (λ x, id₁ _ ,, id_pred_mor D _ ,, linvunitor_invertible_2cell _).
- cbn.
refine (λ x y f, _).
use make_invertible_2cell.
+ cbn.
simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite !vassocr ;
rewrite rinvunitor_runitor ;
rewrite id2_left ;
rewrite lwhisker_hcomp ;
rewrite <- linvunitor_natural ;
rewrite !vassocl ;
apply maponpaths ;
rewrite linvunitor_assoc ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocr ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
rewrite !vassocl ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
apply idpath).
+ apply is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
- exact (λ x, id₁ _ ,, id_pred_mor D _ ,, linvunitor_invertible_2cell _).
- cbn.
refine (λ x y f, _).
use make_invertible_2cell.
+ cbn.
simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite !vassocr ;
rewrite rinvunitor_runitor ;
rewrite id2_left ;
rewrite lwhisker_hcomp ;
rewrite <- linvunitor_natural ;
rewrite !vassocl ;
apply maponpaths ;
rewrite linvunitor_assoc ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocr ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
rewrite !vassocl ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
apply idpath).
+ apply is_invertible_2cell_in_disp_map_slice_bicat ; cbn.
is_iso.
Definition to_fiber_disp_map_unit_inv_is_pstrans
: is_pstrans to_fiber_disp_map_unit_inv_data.
Show proof.
repeat split.
- intros x y f g α.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
apply idpath.
- intros x.
use eq_2cell_disp_map_slice.
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
rewrite runitor_lunitor_identity, lunitor_linvunitor.
rewrite lunitor_runitor_identity, runitor_rinvunitor.
apply idpath.
- intros x y z f g.
use eq_2cell_disp_map_slice.
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
use vcomp_move_R_Mp ; [ is_iso | ].
cbn.
rewrite <- !rwhisker_vcomp.
rewrite !vassocr.
rewrite lunitor_triangle.
rewrite !vassocl.
refine (!(id2_right _) @ _).
apply maponpaths.
rewrite <- runitor_triangle.
rewrite (maponpaths (λ z, _ • (_ • (_ • z))) (vassocr _ _ _)).
rewrite lassociator_rassociator.
rewrite id2_left.
rewrite lwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
use vcomp_move_L_pM ; [ is_iso | ] ; cbn.
rewrite id2_right.
rewrite lunitor_lwhisker.
apply idpath.
- intros x y f g α.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
apply idpath.
- intros x.
use eq_2cell_disp_map_slice.
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
rewrite runitor_lunitor_identity, lunitor_linvunitor.
rewrite lunitor_runitor_identity, runitor_rinvunitor.
apply idpath.
- intros x y z f g.
use eq_2cell_disp_map_slice.
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
use vcomp_move_R_Mp ; [ is_iso | ].
cbn.
rewrite <- !rwhisker_vcomp.
rewrite !vassocr.
rewrite lunitor_triangle.
rewrite !vassocl.
refine (!(id2_right _) @ _).
apply maponpaths.
rewrite <- runitor_triangle.
rewrite (maponpaths (λ z, _ • (_ • (_ • z))) (vassocr _ _ _)).
rewrite lassociator_rassociator.
rewrite id2_left.
rewrite lwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
use vcomp_move_L_pM ; [ is_iso | ] ; cbn.
rewrite id2_right.
rewrite lunitor_lwhisker.
apply idpath.
Definition to_fiber_disp_map_unit_inv
: pstrans
(comp_psfunctor from_fiber_disp_map to_fiber_disp_map)
(id_psfunctor _).
Show proof.
use make_pstrans.
- exact to_fiber_disp_map_unit_inv_data.
- exact to_fiber_disp_map_unit_inv_is_pstrans.
- exact to_fiber_disp_map_unit_inv_data.
- exact to_fiber_disp_map_unit_inv_is_pstrans.
5. The counit
Definition to_fiber_disp_map_counit_data
: pstrans_data
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map)
(id_psfunctor _).
Show proof.
Opaque comp_psfunctor.
Definition to_fiber_disp_map_counit_is_pstrans
: is_pstrans to_fiber_disp_map_counit_data.
Show proof.
Transparent comp_psfunctor.
Definition to_fiber_disp_map_counit
: pstrans
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map)
(id_psfunctor _).
Show proof.
Definition to_fiber_disp_map_counit_inv_data
: pstrans_data
(id_psfunctor _)
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map).
Show proof.
Opaque strict_fiber_bicat comp_psfunctor.
Definition to_fiber_disp_map_counit_inv_is_pstrans
: is_pstrans to_fiber_disp_map_counit_inv_data.
Show proof.
Transparent strict_fiber_bicat comp_psfunctor.
Definition to_fiber_disp_map_counit_inv
: pstrans
(id_psfunctor _)
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map).
Show proof.
: pstrans_data
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map)
(id_psfunctor _).
Show proof.
use make_pstrans_data.
- cbn.
refine (λ f, id₁ _ ,, id_pred_mor D _ ,, _).
exact (comp_of_invertible_2cell
(runitor_invertible_2cell _)
(linvunitor_invertible_2cell _)).
- refine (λ x y f, _).
use make_invertible_2cell.
+ simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite <- !rwhisker_vcomp ;
rewrite !vassocl ;
do 3 apply maponpaths ;
rewrite <- !lwhisker_vcomp ;
rewrite !vassocl ;
refine (!_) ;
etrans ;
[ do 4 apply maponpaths ;
rewrite lwhisker_hcomp ;
apply triangle_l_inv
| ] ;
rewrite <- rwhisker_hcomp ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite !vassocl ;
rewrite lunitor_triangle ;
rewrite runitor_triangle ;
rewrite vcomp_lunitor ;
rewrite vcomp_runitor ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite !vassocl ;
rewrite <- lunitor_triangle ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
use vcomp_move_R_pM ; [ is_iso | ] ; cbn ;
rewrite id2_right ;
rewrite <- runitor_triangle ;
rewrite runitor_lunitor_identity ;
apply lunitor_lwhisker).
+ apply strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
- cbn.
refine (λ f, id₁ _ ,, id_pred_mor D _ ,, _).
exact (comp_of_invertible_2cell
(runitor_invertible_2cell _)
(linvunitor_invertible_2cell _)).
- refine (λ x y f, _).
use make_invertible_2cell.
+ simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite <- !rwhisker_vcomp ;
rewrite !vassocl ;
do 3 apply maponpaths ;
rewrite <- !lwhisker_vcomp ;
rewrite !vassocl ;
refine (!_) ;
etrans ;
[ do 4 apply maponpaths ;
rewrite lwhisker_hcomp ;
apply triangle_l_inv
| ] ;
rewrite <- rwhisker_hcomp ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite !vassocl ;
rewrite lunitor_triangle ;
rewrite runitor_triangle ;
rewrite vcomp_lunitor ;
rewrite vcomp_runitor ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite !vassocl ;
rewrite <- lunitor_triangle ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
use vcomp_move_R_pM ; [ is_iso | ] ; cbn ;
rewrite id2_right ;
rewrite <- runitor_triangle ;
rewrite runitor_lunitor_identity ;
apply lunitor_lwhisker).
+ apply strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
Opaque comp_psfunctor.
Definition to_fiber_disp_map_counit_is_pstrans
: is_pstrans to_fiber_disp_map_counit_data.
Show proof.
refine (_ ,, _ ,, _).
- intros x y f g α.
use path_2cell_disp_map_fiber.
rewrite !comp_disp_map_fiber.
rewrite lwhisker_disp_map_fiber.
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
apply idpath.
- intros x.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply lwhisker_disp_map_fiber.
}
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply runitor_disp_map_fiber.
}
apply maponpaths.
apply linvunitor_disp_map_fiber.
}
etrans.
{
apply maponpaths.
etrans.
{
apply rwhisker_disp_map_fiber.
}
apply maponpaths.
apply comp_disp_map_fiber.
}
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
rewrite lunitor_runitor_identity, runitor_rinvunitor.
rewrite runitor_lunitor_identity, lunitor_linvunitor.
apply idpath.
- intros x y z f g.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths (λ z, z • _) (lwhisker_disp_map_fiber _ _) @ _).
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths
(λ z, z • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (z • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((z • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (((z • _) • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((z • _) • _) • _) • _) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • z) • _) • _) • _) • _)
(rwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • z) • _) • _) • _)
(rassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • z) • _) • _)
(lwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • z) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • _) • z)
(rwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • _) • (z ▹ _))
(comp_disp_map_fiber _ _) @ _).
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
rewrite <- rwhisker_vcomp.
rewrite <- lunitor_triangle.
rewrite !vassocl.
do 2 apply maponpaths.
rewrite left_unit_inv_assoc₂.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
apply maponpaths_2.
refine (_ @ id2_left _).
apply maponpaths_2.
rewrite !vassocl.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite rinvunitor_runitor.
rewrite id2_rwhisker.
apply idpath.
- intros x y f g α.
use path_2cell_disp_map_fiber.
rewrite !comp_disp_map_fiber.
rewrite lwhisker_disp_map_fiber.
rewrite rwhisker_disp_map_fiber.
cbn.
rewrite !vassocr.
rewrite vcomp_lunitor.
rewrite !vassocl.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
apply idpath.
- intros x.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply lwhisker_disp_map_fiber.
}
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply runitor_disp_map_fiber.
}
apply maponpaths.
apply linvunitor_disp_map_fiber.
}
etrans.
{
apply maponpaths.
etrans.
{
apply rwhisker_disp_map_fiber.
}
apply maponpaths.
apply comp_disp_map_fiber.
}
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
rewrite lunitor_runitor_identity, runitor_rinvunitor.
rewrite runitor_lunitor_identity, lunitor_linvunitor.
apply idpath.
- intros x y z f g.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths (λ z, z • _) (lwhisker_disp_map_fiber _ _) @ _).
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths
(λ z, z • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (z • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((z • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (((z • _) • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((z • _) • _) • _) • _) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • z) • _) • _) • _) • _)
(rwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • z) • _) • _) • _)
(rassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • z) • _) • _)
(lwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • z) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • _) • z)
(rwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • _) • (z ▹ _))
(comp_disp_map_fiber _ _) @ _).
cbn.
rewrite lwhisker_id2, !id2_left.
rewrite id2_rwhisker, id2_right.
rewrite <- rwhisker_vcomp.
rewrite <- lunitor_triangle.
rewrite !vassocl.
do 2 apply maponpaths.
rewrite left_unit_inv_assoc₂.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
apply maponpaths_2.
refine (_ @ id2_left _).
apply maponpaths_2.
rewrite !vassocl.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite rinvunitor_runitor.
rewrite id2_rwhisker.
apply idpath.
Transparent comp_psfunctor.
Definition to_fiber_disp_map_counit
: pstrans
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map)
(id_psfunctor _).
Show proof.
use make_pstrans.
- exact to_fiber_disp_map_counit_data.
- exact to_fiber_disp_map_counit_is_pstrans.
- exact to_fiber_disp_map_counit_data.
- exact to_fiber_disp_map_counit_is_pstrans.
Definition to_fiber_disp_map_counit_inv_data
: pstrans_data
(id_psfunctor _)
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map).
Show proof.
use make_pstrans_data.
- refine (λ x, id₁ _ ,, id_pred_mor D _ ,, _).
exact (comp_of_invertible_2cell
(runitor_invertible_2cell _)
(linvunitor_invertible_2cell _)).
- intros x y f.
use make_invertible_2cell.
+ simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite <- !rwhisker_vcomp ;
rewrite !vassocl ;
do 2 apply maponpaths ;
rewrite !vassocr ;
rewrite runitor_rinvunitor ;
rewrite id2_left ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite !vassocl ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite !vassocl ;
rewrite linvunitor_lunitor ;
rewrite id2_right ;
rewrite runitor_triangle ;
rewrite lunitor_triangle ;
rewrite vcomp_lunitor ;
rewrite vcomp_runitor ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite !vassocl ;
rewrite <- lunitor_triangle ;
rewrite (maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
rewrite id2_right ;
rewrite <- runitor_triangle ;
rewrite runitor_lunitor_identity ;
rewrite lunitor_lwhisker ;
apply idpath).
+ apply strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
- refine (λ x, id₁ _ ,, id_pred_mor D _ ,, _).
exact (comp_of_invertible_2cell
(runitor_invertible_2cell _)
(linvunitor_invertible_2cell _)).
- intros x y f.
use make_invertible_2cell.
+ simple refine (_ ,, _).
* cbn.
exact (lunitor _ • rinvunitor _).
* abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite <- !rwhisker_vcomp ;
rewrite !vassocl ;
do 2 apply maponpaths ;
rewrite !vassocr ;
rewrite runitor_rinvunitor ;
rewrite id2_left ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite !vassocl ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite !vassocl ;
rewrite linvunitor_lunitor ;
rewrite id2_right ;
rewrite runitor_triangle ;
rewrite lunitor_triangle ;
rewrite vcomp_lunitor ;
rewrite vcomp_runitor ;
rewrite !vassocr ;
apply maponpaths_2 ;
rewrite !vassocl ;
rewrite <- lunitor_triangle ;
rewrite (maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)) ;
rewrite rassociator_lassociator ;
rewrite id2_left ;
rewrite rwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite id2_rwhisker ;
rewrite id2_right ;
rewrite <- runitor_triangle ;
rewrite runitor_lunitor_identity ;
rewrite lunitor_lwhisker ;
apply idpath).
+ apply strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
Opaque strict_fiber_bicat comp_psfunctor.
Definition to_fiber_disp_map_counit_inv_is_pstrans
: is_pstrans to_fiber_disp_map_counit_inv_data.
Show proof.
repeat split.
- intros x y f g α.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _ @ !(comp_disp_map_fiber _ _)).
etrans.
{
apply maponpaths_2.
apply lwhisker_disp_map_fiber.
}
refine (!_).
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
cbn.
refine (!_).
refine (vassocr _ _ _ @ _).
etrans.
{
apply maponpaths_2.
apply vcomp_lunitor.
}
refine (vassocl _ _ _ @ _ @ vassocr _ _ _).
apply maponpaths.
etrans.
{
apply rinvunitor_natural.
}
apply maponpaths.
refine (!_).
apply rwhisker_hcomp.
- intros x.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
etrans.
{
apply lwhisker_disp_map_fiber.
}
apply maponpaths.
exact (comp_disp_map_fiber
(psfunctor_id to_fiber_disp_map x)
(##to_fiber_disp_map (psfunctor_id from_fiber_disp_map x))).
}
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply runitor_disp_map_fiber.
}
apply maponpaths.
apply linvunitor_disp_map_fiber.
}
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
cbn.
etrans.
{
apply maponpaths.
apply id2_rwhisker.
}
refine (id2_right _ @ _).
refine (!_).
etrans.
{
apply maponpaths_2.
etrans.
{
apply maponpaths.
apply id2_left.
}
apply lwhisker_id2.
}
refine (id2_left _ @ _).
etrans.
{
apply maponpaths_2.
apply lunitor_runitor_identity.
}
refine (runitor_rinvunitor _ @ _).
refine (!_).
etrans.
{
apply maponpaths_2.
apply runitor_lunitor_identity.
}
apply lunitor_linvunitor.
- intros x y z f g.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths (λ z, z • _) (lwhisker_disp_map_fiber _ _) @ _).
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths
(λ z, z • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (z • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((z • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (((z • _) • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((z • _) • _) • _) • _) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • z) • _) • _) • _) • _)
(rwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • z) • _) • _) • _)
(rassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • z) • _) • _)
(lwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • z) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • _) • z)
(rwhisker_disp_map_fiber _ _) @ _).
refine (!_).
etrans.
{
apply maponpaths_2.
apply maponpaths.
exact (comp_disp_map_fiber
(psfunctor_comp
to_fiber_disp_map
(# from_fiber_disp_map f)
(# from_fiber_disp_map g))
(## to_fiber_disp_map (psfunctor_comp from_fiber_disp_map f g))).
}
cbn.
etrans.
{
apply maponpaths_2.
etrans.
{
apply maponpaths.
apply id2_left.
}
apply lwhisker_id2.
}
refine (id2_left _ @ _).
refine (!_).
etrans.
{
apply maponpaths.
apply id2_rwhisker.
}
refine (id2_right _ @ _).
etrans.
{
do 3 apply maponpaths_2.
etrans.
{
apply maponpaths.
refine (!_).
apply rwhisker_vcomp.
}
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
apply lunitor_triangle.
}
do 3 refine (vassocl _ _ _ @ _).
apply maponpaths.
etrans.
{
do 2 apply maponpaths.
etrans.
{
apply maponpaths_2.
refine (!_).
apply lwhisker_vcomp.
}
refine (vassocl _ _ _ @ _).
apply maponpaths.
etrans.
{
apply maponpaths_2.
apply left_unit_inv_assoc.
}
refine (vassocl _ _ _ @ _).
etrans.
{
apply maponpaths.
apply rassociator_lassociator.
}
apply id2_right.
}
etrans.
{
apply maponpaths.
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
apply lunitor_lwhisker.
}
refine (vassocr _ _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (rwhisker_vcomp _ _ _ @ _).
etrans.
{
apply maponpaths.
apply rinvunitor_runitor.
}
apply id2_rwhisker.
}
apply id2_left.
- intros x y f g α.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _ @ !(comp_disp_map_fiber _ _)).
etrans.
{
apply maponpaths_2.
apply lwhisker_disp_map_fiber.
}
refine (!_).
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
cbn.
refine (!_).
refine (vassocr _ _ _ @ _).
etrans.
{
apply maponpaths_2.
apply vcomp_lunitor.
}
refine (vassocl _ _ _ @ _ @ vassocr _ _ _).
apply maponpaths.
etrans.
{
apply rinvunitor_natural.
}
apply maponpaths.
refine (!_).
apply rwhisker_hcomp.
- intros x.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
etrans.
{
apply lwhisker_disp_map_fiber.
}
apply maponpaths.
exact (comp_disp_map_fiber
(psfunctor_id to_fiber_disp_map x)
(##to_fiber_disp_map (psfunctor_id from_fiber_disp_map x))).
}
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply runitor_disp_map_fiber.
}
apply maponpaths.
apply linvunitor_disp_map_fiber.
}
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
cbn.
etrans.
{
apply maponpaths.
apply id2_rwhisker.
}
refine (id2_right _ @ _).
refine (!_).
etrans.
{
apply maponpaths_2.
etrans.
{
apply maponpaths.
apply id2_left.
}
apply lwhisker_id2.
}
refine (id2_left _ @ _).
etrans.
{
apply maponpaths_2.
apply lunitor_runitor_identity.
}
refine (runitor_rinvunitor _ @ _).
refine (!_).
etrans.
{
apply maponpaths_2.
apply runitor_lunitor_identity.
}
apply lunitor_linvunitor.
- intros x y z f g.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths (λ z, z • _) (lwhisker_disp_map_fiber _ _) @ _).
refine (!_).
refine (comp_disp_map_fiber _ _ @ _).
refine (maponpaths
(λ z, z • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (z • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((z • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, (((z • _) • _) • _) • _)
(comp_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((z • _) • _) • _) • _) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • z) • _) • _) • _) • _)
(rwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • z) • _) • _) • _)
(rassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • z) • _) • _)
(lwhisker_disp_map_fiber _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • z) • _)
(lassociator_disp_map_fiber _ _ _) @ _).
refine (maponpaths
(λ z, ((((_ • _) • _) • _) • _) • z)
(rwhisker_disp_map_fiber _ _) @ _).
refine (!_).
etrans.
{
apply maponpaths_2.
apply maponpaths.
exact (comp_disp_map_fiber
(psfunctor_comp
to_fiber_disp_map
(# from_fiber_disp_map f)
(# from_fiber_disp_map g))
(## to_fiber_disp_map (psfunctor_comp from_fiber_disp_map f g))).
}
cbn.
etrans.
{
apply maponpaths_2.
etrans.
{
apply maponpaths.
apply id2_left.
}
apply lwhisker_id2.
}
refine (id2_left _ @ _).
refine (!_).
etrans.
{
apply maponpaths.
apply id2_rwhisker.
}
refine (id2_right _ @ _).
etrans.
{
do 3 apply maponpaths_2.
etrans.
{
apply maponpaths.
refine (!_).
apply rwhisker_vcomp.
}
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
apply lunitor_triangle.
}
do 3 refine (vassocl _ _ _ @ _).
apply maponpaths.
etrans.
{
do 2 apply maponpaths.
etrans.
{
apply maponpaths_2.
refine (!_).
apply lwhisker_vcomp.
}
refine (vassocl _ _ _ @ _).
apply maponpaths.
etrans.
{
apply maponpaths_2.
apply left_unit_inv_assoc.
}
refine (vassocl _ _ _ @ _).
etrans.
{
apply maponpaths.
apply rassociator_lassociator.
}
apply id2_right.
}
etrans.
{
apply maponpaths.
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
apply lunitor_lwhisker.
}
refine (vassocr _ _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (rwhisker_vcomp _ _ _ @ _).
etrans.
{
apply maponpaths.
apply rinvunitor_runitor.
}
apply id2_rwhisker.
}
apply id2_left.
Transparent strict_fiber_bicat comp_psfunctor.
Definition to_fiber_disp_map_counit_inv
: pstrans
(id_psfunctor _)
(comp_psfunctor to_fiber_disp_map from_fiber_disp_map).
Show proof.
use make_pstrans.
- exact to_fiber_disp_map_counit_inv_data.
- exact to_fiber_disp_map_counit_inv_is_pstrans.
- exact to_fiber_disp_map_counit_inv_data.
- exact to_fiber_disp_map_counit_inv_is_pstrans.
6. The modifications
Definition disp_map_unit_inv_left_data
: invertible_modification_data
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_unit_inv to_fiber_disp_map_unit).
Show proof.
Definition disp_map_unit_inv_left_is_modification
: is_modification disp_map_unit_inv_left_data.
Show proof.
Definition disp_map_unit_inv_left
: invertible_modification
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_unit_inv to_fiber_disp_map_unit).
Show proof.
Definition disp_map_unit_inv_right_data
: invertible_modification_data
(comp_pstrans to_fiber_disp_map_unit to_fiber_disp_map_unit_inv)
(id_pstrans _).
Show proof.
Definition disp_map_unit_inv_right_is_modification
: is_modification disp_map_unit_inv_right_data.
Show proof.
Definition disp_map_unit_inv_right
: invertible_modification
(comp_pstrans to_fiber_disp_map_unit to_fiber_disp_map_unit_inv)
(id_pstrans _).
Show proof.
Definition disp_map_counit_inv_right_data
: invertible_modification_data
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_counit to_fiber_disp_map_counit_inv).
Show proof.
Opaque comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_right_is_modification
: is_modification disp_map_counit_inv_right_data.
Show proof.
Transparent comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_right
: invertible_modification
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_counit to_fiber_disp_map_counit_inv).
Show proof.
Definition disp_map_counit_inv_left_data
: invertible_modification_data
(comp_pstrans to_fiber_disp_map_counit_inv to_fiber_disp_map_counit)
(id_pstrans _).
Show proof.
Opaque comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_left_is_modification
: is_modification disp_map_counit_inv_left_data.
Show proof.
Transparent comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_left
: invertible_modification
(comp_pstrans to_fiber_disp_map_counit_inv to_fiber_disp_map_counit)
(id_pstrans _).
Show proof.
: invertible_modification_data
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_unit_inv to_fiber_disp_map_unit).
Show proof.
intros x.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (linvunitor _).
+ abstract
(cbn ;
apply maponpaths ;
use vcomp_move_L_pM ; [ is_iso | ] ; cbn ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite lunitor_runitor_identity ;
rewrite lwhisker_hcomp, rwhisker_hcomp ;
apply triangle_r).
- apply is_invertible_2cell_in_disp_map_slice_bicat.
cbn.
is_iso.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (linvunitor _).
+ abstract
(cbn ;
apply maponpaths ;
use vcomp_move_L_pM ; [ is_iso | ] ; cbn ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite lunitor_runitor_identity ;
rewrite lwhisker_hcomp, rwhisker_hcomp ;
apply triangle_r).
- apply is_invertible_2cell_in_disp_map_slice_bicat.
cbn.
is_iso.
Definition disp_map_unit_inv_left_is_modification
: is_modification disp_map_unit_inv_left_data.
Show proof.
intros x y f.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocr.
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn.
rewrite !vassocl.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite <- lwhisker_vcomp.
refine (!_).
etrans.
{
apply maponpaths.
rewrite !vassocr.
rewrite lunitor_lwhisker.
apply idpath.
}
rewrite runitor_lunitor_identity.
rewrite !vassocr.
rewrite rwhisker_vcomp.
rewrite linvunitor_lunitor.
rewrite id2_rwhisker.
rewrite id2_left.
rewrite !vassocl.
rewrite lunitor_triangle.
rewrite vcomp_lunitor.
apply idpath.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocr.
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn.
rewrite !vassocl.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite <- lwhisker_vcomp.
refine (!_).
etrans.
{
apply maponpaths.
rewrite !vassocr.
rewrite lunitor_lwhisker.
apply idpath.
}
rewrite runitor_lunitor_identity.
rewrite !vassocr.
rewrite rwhisker_vcomp.
rewrite linvunitor_lunitor.
rewrite id2_rwhisker.
rewrite id2_left.
rewrite !vassocl.
rewrite lunitor_triangle.
rewrite vcomp_lunitor.
apply idpath.
Definition disp_map_unit_inv_left
: invertible_modification
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_unit_inv to_fiber_disp_map_unit).
Show proof.
use make_invertible_modification.
- exact disp_map_unit_inv_left_data.
- exact disp_map_unit_inv_left_is_modification.
- exact disp_map_unit_inv_left_data.
- exact disp_map_unit_inv_left_is_modification.
Definition disp_map_unit_inv_right_data
: invertible_modification_data
(comp_pstrans to_fiber_disp_map_unit to_fiber_disp_map_unit_inv)
(id_pstrans _).
Show proof.
intros x.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (lunitor _).
+ abstract
(cbn ;
refine (_ @ id2_right _) ;
rewrite !vassocl ;
apply maponpaths ;
use vcomp_move_R_pM ; [ is_iso | ] ; cbn ;
rewrite id2_right ;
rewrite lunitor_runitor_identity ;
rewrite lwhisker_hcomp, rwhisker_hcomp ;
refine (!_) ;
apply triangle_r).
- apply is_invertible_2cell_in_disp_map_slice_bicat.
cbn.
is_iso.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (lunitor _).
+ abstract
(cbn ;
refine (_ @ id2_right _) ;
rewrite !vassocl ;
apply maponpaths ;
use vcomp_move_R_pM ; [ is_iso | ] ; cbn ;
rewrite id2_right ;
rewrite lunitor_runitor_identity ;
rewrite lwhisker_hcomp, rwhisker_hcomp ;
refine (!_) ;
apply triangle_r).
- apply is_invertible_2cell_in_disp_map_slice_bicat.
cbn.
is_iso.
Definition disp_map_unit_inv_right_is_modification
: is_modification disp_map_unit_inv_right_data.
Show proof.
intros x y f.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocl.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite <- lwhisker_vcomp.
rewrite lunitor_triangle.
use vcomp_move_R_pM ; [ is_iso | ] ; cbn.
rewrite !vassocr.
rewrite lunitor_triangle.
rewrite lwhisker_vcomp.
rewrite vcomp_lunitor.
rewrite !vassocl.
apply idpath.
use eq_2cell_disp_map_slice.
cbn.
rewrite !vassocl.
rewrite lunitor_lwhisker.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor.
rewrite id2_right.
rewrite <- lwhisker_vcomp.
rewrite lunitor_triangle.
use vcomp_move_R_pM ; [ is_iso | ] ; cbn.
rewrite !vassocr.
rewrite lunitor_triangle.
rewrite lwhisker_vcomp.
rewrite vcomp_lunitor.
rewrite !vassocl.
apply idpath.
Definition disp_map_unit_inv_right
: invertible_modification
(comp_pstrans to_fiber_disp_map_unit to_fiber_disp_map_unit_inv)
(id_pstrans _).
Show proof.
use make_invertible_modification.
- exact disp_map_unit_inv_right_data.
- exact disp_map_unit_inv_right_is_modification.
- exact disp_map_unit_inv_right_data.
- exact disp_map_unit_inv_right_is_modification.
Definition disp_map_counit_inv_right_data
: invertible_modification_data
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_counit to_fiber_disp_map_counit_inv).
Show proof.
intros x.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (linvunitor _).
+ abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite !vassocr ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite !vassocl ;
rewrite lunitor_runitor_identity ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite !vassocl ;
rewrite linvunitor_lunitor ;
rewrite id2_right ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite <- lunitor_lwhisker ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite lassociator_rassociator ;
rewrite id2_left ;
rewrite !vassocr ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
rewrite !vassocl ;
rewrite runitor_triangle ;
rewrite vcomp_runitor ;
apply idpath).
- use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (linvunitor _).
+ abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite !vassocr ;
use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ;
rewrite !vassocl ;
rewrite lunitor_runitor_identity ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite !vassocl ;
rewrite linvunitor_lunitor ;
rewrite id2_right ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite <- lunitor_lwhisker ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite lassociator_rassociator ;
rewrite id2_left ;
rewrite !vassocr ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
rewrite !vassocl ;
rewrite runitor_triangle ;
rewrite vcomp_runitor ;
apply idpath).
- use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
Opaque comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_right_is_modification
: is_modification disp_map_counit_inv_right_data.
Show proof.
intros x y f.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _ @ !(comp_disp_map_fiber _ _)).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths_2.
cbn.
apply comp_disp_map_fiber.
}
etrans.
{
apply maponpaths_2.
etrans.
{
apply maponpaths_2.
apply lunitor_disp_map_fiber.
}
apply maponpaths.
apply rinvunitor_disp_map_fiber.
}
refine (!_).
etrans.
{
apply maponpaths_2.
apply rwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
apply maponpaths_2.
apply rassociator_disp_map_fiber.
}
cbn.
rewrite !vassocl.
etrans.
{
apply maponpaths.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
do 4 apply maponpaths_2.
apply lunitor_lwhisker.
}
etrans.
{
rewrite !vassocr.
do 3 apply maponpaths_2.
etrans.
{
apply maponpaths_2.
etrans.
{
apply rwhisker_vcomp.
}
apply maponpaths.
etrans.
{
apply maponpaths.
apply runitor_lunitor_identity.
}
apply linvunitor_lunitor.
}
etrans.
{
apply maponpaths_2.
apply id2_rwhisker.
}
apply id2_left.
}
refine (!_).
etrans.
{
refine (vassocr _ _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (!_).
apply vcomp_lunitor.
}
apply vassocl.
}
do 2 refine (_ @ vassocr _ _ _).
apply maponpaths.
refine (!_).
etrans.
{
etrans.
{
apply maponpaths.
apply maponpaths_2.
refine (!_).
apply rwhisker_vcomp.
}
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
apply lunitor_triangle.
}
refine (vassocl _ _ _ @ _).
apply maponpaths.
use vcomp_move_R_pM ; [ is_iso | ].
use vcomp_move_L_Mp ; [ is_iso | ].
apply lunitor_lwhisker.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _ @ !(comp_disp_map_fiber _ _)).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths_2.
cbn.
apply comp_disp_map_fiber.
}
etrans.
{
apply maponpaths_2.
etrans.
{
apply maponpaths_2.
apply lunitor_disp_map_fiber.
}
apply maponpaths.
apply rinvunitor_disp_map_fiber.
}
refine (!_).
etrans.
{
apply maponpaths_2.
apply rwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
apply maponpaths_2.
apply rassociator_disp_map_fiber.
}
cbn.
rewrite !vassocl.
etrans.
{
apply maponpaths.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
do 4 apply maponpaths_2.
apply lunitor_lwhisker.
}
etrans.
{
rewrite !vassocr.
do 3 apply maponpaths_2.
etrans.
{
apply maponpaths_2.
etrans.
{
apply rwhisker_vcomp.
}
apply maponpaths.
etrans.
{
apply maponpaths.
apply runitor_lunitor_identity.
}
apply linvunitor_lunitor.
}
etrans.
{
apply maponpaths_2.
apply id2_rwhisker.
}
apply id2_left.
}
refine (!_).
etrans.
{
refine (vassocr _ _ _ @ _).
etrans.
{
apply maponpaths_2.
refine (!_).
apply vcomp_lunitor.
}
apply vassocl.
}
do 2 refine (_ @ vassocr _ _ _).
apply maponpaths.
refine (!_).
etrans.
{
etrans.
{
apply maponpaths.
apply maponpaths_2.
refine (!_).
apply rwhisker_vcomp.
}
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
apply lunitor_triangle.
}
refine (vassocl _ _ _ @ _).
apply maponpaths.
use vcomp_move_R_pM ; [ is_iso | ].
use vcomp_move_L_Mp ; [ is_iso | ].
apply lunitor_lwhisker.
Transparent comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_right
: invertible_modification
(id_pstrans _)
(comp_pstrans to_fiber_disp_map_counit to_fiber_disp_map_counit_inv).
Show proof.
use make_invertible_modification.
- exact disp_map_counit_inv_right_data.
- exact disp_map_counit_inv_right_is_modification.
- exact disp_map_counit_inv_right_data.
- exact disp_map_counit_inv_right_is_modification.
Definition disp_map_counit_inv_left_data
: invertible_modification_data
(comp_pstrans to_fiber_disp_map_counit_inv to_fiber_disp_map_counit)
(id_pstrans _).
Show proof.
intros x.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (lunitor _).
+ abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite !vassocl ;
rewrite lunitor_runitor_identity ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite !vassocl ;
rewrite linvunitor_lunitor ;
rewrite id2_right ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite <- lunitor_lwhisker ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite lassociator_rassociator ;
rewrite id2_left ;
rewrite !vassocr ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
rewrite !vassocl ;
rewrite runitor_triangle ;
rewrite vcomp_runitor ;
apply idpath).
- use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
use make_invertible_2cell.
- simple refine (_ ,, _).
+ exact (lunitor _).
+ abstract
(cbn ;
rewrite lwhisker_id2, id2_left ;
rewrite !vassocl ;
rewrite lunitor_runitor_identity ;
rewrite runitor_rwhisker ;
rewrite lwhisker_vcomp ;
rewrite !vassocl ;
rewrite linvunitor_lunitor ;
rewrite id2_right ;
rewrite <- rwhisker_vcomp ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite <- lunitor_lwhisker ;
rewrite !vassocl ;
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ;
rewrite lassociator_rassociator ;
rewrite id2_left ;
rewrite !vassocr ;
rewrite lwhisker_vcomp ;
rewrite linvunitor_lunitor ;
rewrite lwhisker_id2 ;
rewrite id2_left ;
rewrite !vassocl ;
rewrite runitor_triangle ;
rewrite vcomp_runitor ;
apply idpath).
- use strict_fiber_bicat_invertible_2cell.
use is_invertible_to_is_disp_invertible.
cbn.
is_iso.
Opaque comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_left_is_modification
: is_modification disp_map_counit_inv_left_data.
Show proof.
intros x y f.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _ @ !(comp_disp_map_fiber _ _)).
refine (!_).
etrans.
{
apply maponpaths_2.
apply rwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply lunitor_disp_map_fiber.
}
apply maponpaths.
apply rinvunitor_disp_map_fiber.
}
refine (!_).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
apply maponpaths_2.
apply rassociator_disp_map_fiber.
}
cbn.
etrans.
{
do 4 apply maponpaths_2.
etrans.
{
apply maponpaths.
refine (!_).
apply lwhisker_vcomp.
}
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
etrans.
{
apply lunitor_lwhisker.
}
apply maponpaths.
apply runitor_lunitor_identity.
}
do 4 refine (vassocl _ _ _ @ _).
apply maponpaths.
etrans.
{
do 2 apply maponpaths.
etrans.
{
apply maponpaths.
apply lunitor_lwhisker.
}
refine (rwhisker_vcomp _ _ _ @ _).
cbn.
apply maponpaths.
refine (vassocl _ _ _ @ _).
etrans.
{
apply maponpaths.
apply rinvunitor_runitor.
}
apply id2_right.
}
etrans.
{
apply maponpaths.
apply lunitor_triangle.
}
apply vcomp_lunitor.
use path_2cell_disp_map_fiber.
refine (comp_disp_map_fiber _ _ @ _ @ !(comp_disp_map_fiber _ _)).
refine (!_).
etrans.
{
apply maponpaths_2.
apply rwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths_2.
apply lunitor_disp_map_fiber.
}
apply maponpaths.
apply rinvunitor_disp_map_fiber.
}
refine (!_).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
etrans.
{
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply rwhisker_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lassociator_disp_map_fiber.
}
apply maponpaths_2.
refine (comp_disp_map_fiber _ _ @ _).
etrans.
{
apply maponpaths.
apply lwhisker_disp_map_fiber.
}
apply maponpaths_2.
apply rassociator_disp_map_fiber.
}
cbn.
etrans.
{
do 4 apply maponpaths_2.
etrans.
{
apply maponpaths.
refine (!_).
apply lwhisker_vcomp.
}
refine (vassocr _ _ _ @ _).
apply maponpaths_2.
etrans.
{
apply lunitor_lwhisker.
}
apply maponpaths.
apply runitor_lunitor_identity.
}
do 4 refine (vassocl _ _ _ @ _).
apply maponpaths.
etrans.
{
do 2 apply maponpaths.
etrans.
{
apply maponpaths.
apply lunitor_lwhisker.
}
refine (rwhisker_vcomp _ _ _ @ _).
cbn.
apply maponpaths.
refine (vassocl _ _ _ @ _).
etrans.
{
apply maponpaths.
apply rinvunitor_runitor.
}
apply id2_right.
}
etrans.
{
apply maponpaths.
apply lunitor_triangle.
}
apply vcomp_lunitor.
Transparent comp_psfunctor strict_fiber_bicat.
Definition disp_map_counit_inv_left
: invertible_modification
(comp_pstrans to_fiber_disp_map_counit_inv to_fiber_disp_map_counit)
(id_pstrans _).
Show proof.
use make_invertible_modification.
- exact disp_map_counit_inv_left_data.
- exact disp_map_counit_inv_left_is_modification.
- exact disp_map_counit_inv_left_data.
- exact disp_map_counit_inv_left_is_modification.
7. The biequivalence
Definition to_fiber_disp_map_is_biequivalence
: is_biequivalence to_fiber_disp_map.
Show proof.
: is_biequivalence to_fiber_disp_map.
Show proof.
use make_is_biequivalence.
- exact from_fiber_disp_map.
- exact to_fiber_disp_map_unit.
- exact to_fiber_disp_map_unit_inv.
- exact to_fiber_disp_map_counit.
- exact to_fiber_disp_map_counit_inv.
- exact disp_map_unit_inv_left.
- exact disp_map_unit_inv_right.
- exact disp_map_counit_inv_right.
- exact disp_map_counit_inv_left.
End FiberOfDisplayMap.- exact from_fiber_disp_map.
- exact to_fiber_disp_map_unit.
- exact to_fiber_disp_map_unit_inv.
- exact to_fiber_disp_map_counit.
- exact to_fiber_disp_map_counit_inv.
- exact disp_map_unit_inv_left.
- exact disp_map_unit_inv_right.
- exact disp_map_counit_inv_right.
- exact disp_map_counit_inv_left.