Library UniMath.Bicategories.PseudoFunctors.Display.Map1Cells
This is the first layer of the construction of the bicategory of pseudofunctors.
To a function of objects, we add an action of 1-cells.
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.PrecategoryBinProduct.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat.
Import DispBicat.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Base.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.Core.Adjunctions.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions.
Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Local Open Scope cat.
Section Map1Cells.
Variable (C D : bicat).
Definition map1cells_disp_cat : disp_cat_ob_mor (ps_base C D).
Proof.
use tpair.
- exact (λ F₀, ∏ (X Y : C), X --> Y → F₀ X --> F₀ Y).
- exact (λ F₀ G₀ F₁ G₁ η, ∏ (X Y : C) (f : X --> Y),
invertible_2cell (η X · G₁ X Y f) (F₁ X Y f · η Y)).
Defined.
Definition map1cells_disp_cat_id_comp : disp_cat_id_comp (ps_base C D) map1cells_disp_cat.
Proof.
use tpair.
- cbn.
refine (λ F₀ F₁ X Y f, (lunitor (F₁ X Y f) • rinvunitor (F₁ X Y f) ,, _)).
is_iso.
- cbn.
refine (λ F₀ G₀ H₀ η₁ ε₁ F₁ G₁ H₁ η₂ ε₂ X Y f,
(rassociator (η₁ X) (ε₁ X) (H₁ X Y f))
• (η₁ X ◃ ε₂ X Y f)
• lassociator (η₁ X) (G₁ X Y f) (ε₁ Y)
• (η₂ X Y f ▹ ε₁ Y)
• rassociator (F₁ X Y f) (η₁ Y) (ε₁ Y) ,, _).
is_iso.
+ apply ε₂.
+ apply η₂.
Defined.
Definition map1cells_disp_cat_2cell : disp_2cell_struct map1cells_disp_cat
:= λ F₀ G₀ η₁ ε₁ m F₁ G₁ η₂ ε₂,
∏ (X Y : C) (f : X --> Y),
η₂ X Y f • (F₁ X Y f ◃ m Y)
=
(m X ▹ G₁ X Y f) • ε₂ X Y f.
Definition map1cells_prebicat_1 : disp_prebicat_1_id_comp_cells (ps_base C D).
Proof.
use tpair.
- use tpair.
+ exact map1cells_disp_cat.
+ exact map1cells_disp_cat_id_comp.
- exact (λ F₀ G₀ η₁ ε₁ m F₁ G₁ η₂ ε₂,
∏ (X Y : C) (f : X --> Y),
η₂ X Y f • (F₁ X Y f ◃ m Y)
=
(m X ▹ G₁ X Y f) • ε₂ X Y f).
Defined.
Definition map1cells_ops : disp_prebicat_ops map1cells_prebicat_1.
Proof.
repeat split.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite lwhisker_id2, id2_right.
rewrite id2_rwhisker, id2_left.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocl.
rewrite (lwhisker_hcomp _ (lunitor _)).
rewrite triangle_l.
rewrite <- !rwhisker_hcomp.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor, id2_right.
rewrite lunitor_triangle.
rewrite vcomp_lunitor.
use vcomp_move_R_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
rewrite <- lunitor_assoc.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocl.
use vcomp_move_R_pM.
{ is_iso. }
cbn.
refine (_ @ vassocl _ _ _).
rewrite !runitor_triangle.
rewrite (rwhisker_hcomp _ (runitor _)).
rewrite <- triangle_r.
rewrite vcomp_runitor.
rewrite <- lwhisker_vcomp, <- lwhisker_hcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite rinvunitor_triangle.
rewrite rinvunitor_runitor, id2_left.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocr.
use vcomp_move_L_Mp.
{ is_iso. }
cbn.
refine (vassocl _ _ _ @ _).
rewrite <- linvunitor_assoc.
rewrite lwhisker_hcomp.
rewrite triangle_l_inv, <- rwhisker_hcomp.
rewrite <- rwhisker_vcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite lunitor_triangle.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite vcomp_lunitor.
rewrite !vassocr.
rewrite linvunitor_lunitor, id2_left.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocr.
use vcomp_move_L_Mp.
{ is_iso. }
cbn.
refine (vassocl _ _ _ @ _).
rewrite rinvunitor_triangle.
rewrite (rwhisker_hcomp _ (rinvunitor _)).
rewrite <- triangle_r_inv.
rewrite <- lwhisker_hcomp.
rewrite lwhisker_vcomp.
rewrite !vassocr.
rewrite linvunitor_lunitor, id2_left.
rewrite rinvunitor_triangle.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
reflexivity.
- intros F₀ G₀ H₀ K₀ α₁ η₁ ε₁ F₁ G₁ H₁ K₁ α₂ η₂ ε₂ X Y f ; cbn in ×.
rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp.
rewrite !vassocl.
refine (!(_ @ _)).
{
rewrite !vassocr.
do 7 apply maponpaths_2.
symmetry.
rewrite lwhisker_hcomp, rwhisker_hcomp.
rewrite vassocl.
apply inverse_pentagon.
}
rewrite !vassocl.
apply maponpaths.
rewrite !vassocr.
rewrite lwhisker_lwhisker_rassociator.
rewrite !vassocl.
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 5 apply maponpaths_2.
rewrite lwhisker_hcomp.
rewrite vassocl, <- inverse_pentagon_6.
rewrite <- rwhisker_hcomp.
reflexivity.
}
rewrite !vassocl.
apply maponpaths.
etrans.
{
rewrite !vassocr.
rewrite rwhisker_lwhisker_rassociator.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 3 apply maponpaths_2.
rewrite !vassocl.
rewrite lwhisker_hcomp, rwhisker_hcomp.
symmetry.
apply inverse_pentagon.
}
rewrite (lwhisker_hcomp _ (rassociator _ _ _)), (rwhisker_hcomp _ (rassociator _ _ _)).
rewrite <- inverse_pentagon.
rewrite !vassocr.
apply maponpaths_2.
rewrite rwhisker_rwhisker_alt.
apply maponpaths_2.
rewrite !vassocl.
rewrite rassociator_lassociator, id2_right.
reflexivity.
- intros F₀ G₀ H₀ K₀ α₁ η₁ ε₁ F₁ G₁ H₁ K₁ α₂ η₂ ε₂ X Y f ; cbn in ×.
rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 8 apply maponpaths_2.
rewrite lwhisker_hcomp, rwhisker_hcomp.
symmetry.
rewrite !vassocl.
apply inverse_pentagon.
}
rewrite !vassocl.
apply maponpaths.
etrans.
{
rewrite !vassocr.
rewrite lwhisker_lwhisker_rassociator.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
rewrite inverse_pentagon.
rewrite <- lwhisker_hcomp, <- rwhisker_hcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite lwhisker_vcomp.
rewrite rassociator_lassociator.
rewrite lwhisker_id2, id2_left.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
etrans.
{
rewrite !vassocr.
rewrite rwhisker_lwhisker_rassociator.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 4 apply maponpaths_2.
rewrite vassocl, lwhisker_hcomp, rwhisker_hcomp.
symmetry.
apply inverse_pentagon.
}
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite rassociator_lassociator, id2_left.
rewrite !vassocr.
rewrite <- rwhisker_rwhisker_alt.
rewrite !vassocl.
apply maponpaths.
symmetry.
rewrite lwhisker_hcomp, rwhisker_hcomp.
apply inverse_pentagon_6.
- intros F₀ G₀ α₁ η₁ ε₁ m₂ n₂ F₁ G₁ α₂ η₂ ε₂ m₃ n₃ X Y f ; cbn in ×.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
rewrite m₃.
rewrite !vassocl.
rewrite n₃.
rewrite !vassocr.
rewrite rwhisker_vcomp.
reflexivity.
- intros F₀ G₀ H₀ α₁ η₁ ε₁ m₂ F₁ G₁ H₁ α₂ η₂ ε₂ m₃ X Y f ; cbn in ×.
rewrite !vassocr.
rewrite <- rwhisker_lwhisker_rassociator.
rewrite !vassocl.
apply maponpaths.
rewrite lwhisker_lwhisker_rassociator.
rewrite !vassocr.
apply maponpaths_2.
rewrite lwhisker_vcomp.
rewrite <- m₃.
rewrite <- lwhisker_vcomp.
rewrite !vassocl.
apply maponpaths.
rewrite vcomp_whisker.
rewrite !vassocr.
rewrite lwhisker_lwhisker.
reflexivity.
- intros F₀ G₀ H₀ α₁ η₁ ε₁ m₂ F₁ G₁ H₁ α₂ η₂ ε₂ m₃ X Y f ; cbn in ×.
rewrite !vassocr.
rewrite rwhisker_rwhisker_alt.
rewrite !vassocl.
apply maponpaths.
rewrite rwhisker_lwhisker_rassociator.
rewrite !vassocr.
apply maponpaths_2.
rewrite vcomp_whisker.
rewrite !vassocl.
apply maponpaths.
rewrite rwhisker_vcomp.
rewrite m₃.
rewrite <- rwhisker_vcomp.
rewrite !vassocr.
rewrite <- rwhisker_rwhisker.
reflexivity.
Qed.
Definition map1cells_ops_laws : disp_prebicat_laws (_ ,, map1cells_ops).
Proof.
repeat split ; intro ; intros ; do 3 (apply funextsec ; intro) ; apply D.
Qed.
Definition map1cells_disp_prebicat : disp_prebicat (ps_base C D)
:= (_ ,, map1cells_ops_laws).
Definition map1cells_disp_bicat : disp_bicat (ps_base C D).
Proof.
refine (map1cells_disp_prebicat ,, _).
intros X Y f g α hX hY hf hg hα hβ.
apply isasetaprop.
do 3 (apply impred ; intro).
apply D.
Defined.
Definition map1cells_disp_univalent_2_1
: disp_univalent_2_1 map1cells_disp_bicat.
Proof.
apply fiberwise_local_univalent_is_univalent_2_1.
intros F G η F₁ G₁ η₁ η₁'.
use isweqimplimpl.
- intro m ; cbn in ×.
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
pose (pr1 m X Y f) as n.
cbn in n.
rewrite id2_rwhisker, lwhisker_id2 in n.
rewrite id2_left, id2_right in n.
apply subtypePath.
+ intro.
apply isaprop_is_invertible_2cell.
+ apply n.
- repeat (apply impred_isaset ; intro).
use isaset_total2.
+ apply D.
+ intro.
apply isasetaprop.
apply isaprop_is_invertible_2cell.
- apply isaproptotal2.
+ intro.
apply (@isaprop_is_disp_invertible_2cell (ps_base C D)).
+ intros.
repeat (apply funextsec ; intro).
apply D.
Defined.
Definition all_invertible_map1cells_inv
{F G : ps_base C D}
{η ε : F --> G}
(m : invertible_2cell η ε)
{F₁ : map1cells_disp_bicat F}
{G₁ : map1cells_disp_bicat G}
{η₁ : F₁ -->[ η ] G₁}
{ε₁ : F₁ -->[ ε ] G₁}
(m₁ : η₁ ==>[ m ] ε₁)
: ε₁ ==>[ m^-1 ] η₁.
Proof.
intros X Y f.
use vcomp_move_R_Mp.
{
is_iso.
apply is_invertible_2cell_to_all_is_invertible.
is_iso.
}
rewrite !vassocl.
use vcomp_move_L_pM.
{
is_iso.
apply is_invertible_2cell_to_all_is_invertible.
is_iso.
}
exact (!(m₁ X Y f)).
Qed.
Definition all_invertible_map1cells
{F G : ps_base C D}
{η ε : F --> G}
(m : invertible_2cell η ε)
{F₁ : map1cells_disp_bicat F}
{G₁ : map1cells_disp_bicat G}
{η₁ : F₁ -->[ η ] G₁}
{ε₁ : F₁ -->[ ε ] G₁}
(m₁ : η₁ ==>[ m ] ε₁)
: is_disp_invertible_2cell m m₁.
Proof.
use tpair.
- exact (all_invertible_map1cells_inv m m₁).
- split ; repeat (apply funextsec ; intro) ; apply D.
Qed.
Section AllInvertible2CellToDispAdjEquiv.
Variable (F₀ : ps_base C D)
(F₁ F₁' : map1cells_disp_bicat F₀)
(η : (∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f))).
Local Definition all_invertible_left_adj
: F₁ -->[ internal_adjoint_equivalence_identity F₀] F₁'.
Proof.
intros X Y f ; cbn.
use tpair.
- exact (lunitor _ • (η X Y f)^-1 • rinvunitor _).
- cbn.
is_iso.
Defined.
Local Definition all_invertible_right_adj
: F₁' -->[ left_adjoint_right_adjoint (internal_adjoint_equivalence_identity F₀)] F₁.
Proof.
intros X Y f.
use tpair.
- exact (lunitor _ • η X Y f • rinvunitor _).
- cbn.
is_iso.
apply η.
Defined.
Local Definition all_invertible_unit
: id_disp F₁ ==>[ left_adjoint_unit (internal_adjoint_equivalence_identity F₀)]
all_invertible_left_adj;;all_invertible_right_adj.
Proof.
intros X Y f ; cbn.
rewrite !vassocr.
rewrite <- linvunitor_assoc.
rewrite !lwhisker_hcomp.
rewrite <- linvunitor_natural.
rewrite !vassocl.
apply maponpaths.
rewrite vassocr.
rewrite rinvunitor_natural.
rewrite !vassocl.
apply maponpaths.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite linvunitor_assoc.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite rassociator_lassociator, id2_left.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite rwhisker_vcomp.
rewrite !vassocr.
rewrite linvunitor_lunitor, id2_left.
rewrite <- rwhisker_hcomp, rwhisker_vcomp.
rewrite !vassocr.
rewrite vcomp_rinv, id2_left.
rewrite triangle_r_inv.
rewrite rwhisker_hcomp.
reflexivity.
Qed.
Local Definition all_invertible_counit
: (all_invertible_right_adj;; all_invertible_left_adj)
==>[left_adjoint_counit (internal_adjoint_equivalence_identity F₀)] id_disp F₁'.
Proof.
intros X Y f ; cbn.
rewrite <- !lwhisker_vcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • (_ • z))) (vassocr _ _ _)).
rewrite rinvunitor_triangle.
rewrite rwhisker_hcomp.
rewrite <- rinvunitor_natural.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • (_ • z))) (vassocr _ _ _)).
rewrite <- (vcomp_lunitor (F₁ X Y f)).
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite lwhisker_vcomp.
rewrite vcomp_linv, lwhisker_id2, id2_left.
rewrite !vassocl.
rewrite lunitor_runitor_identity.
rewrite runitor_triangle.
rewrite rinvunitor_runitor, id2_right.
rewrite !vassocr.
do 2 (apply maponpaths_2).
use vcomp_move_R_pM.
{ is_iso. }
cbn.
rewrite lwhisker_hcomp, rwhisker_hcomp.
rewrite triangle_r.
rewrite lunitor_runitor_identity.
reflexivity.
Qed.
Definition all_invertible_2cell_to_disp_adjoint_equivalence
: disp_adjoint_equivalence (internal_adjoint_equivalence_identity F₀) F₁ F₁'.
Proof.
use tpair.
- exact all_invertible_left_adj.
- use tpair.
+ use tpair.
× exact all_invertible_right_adj.
× split.
** exact all_invertible_unit.
** exact all_invertible_counit.
+ split ; split.
× repeat (apply funextsec ; intro).
apply D.
× repeat (apply funextsec ; intro).
apply D.
× apply all_invertible_map1cells.
× apply all_invertible_map1cells.
Defined.
End AllInvertible2CellToDispAdjEquiv.
Definition disp_adjoint_equivalence_to_all_invertible_2cell
(F₀ : ps_base C D)
(F₁ F₁' : map1cells_disp_bicat F₀)
: disp_adjoint_equivalence (internal_adjoint_equivalence_identity F₀) F₁ F₁'
→
(∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f)).
Proof.
intros m X Y f.
use tpair.
- refine (rinvunitor _ • _ • lunitor _).
exact ((pr1 m X Y f)^-1).
- cbn ; is_iso.
Defined.
Definition all_invertible_2cell_is_disp_adjoint_equivalence
(HD_2_1 : is_univalent_2_1 D)
(F₀ : ps_base C D)
(F₁ F₁' : map1cells_disp_bicat F₀)
: (∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f))
≃
disp_adjoint_equivalence (internal_adjoint_equivalence_identity F₀) F₁ F₁'.
Proof.
refine (make_weq (all_invertible_2cell_to_disp_adjoint_equivalence F₀ F₁ F₁') _).
use isweq_iso.
- exact (disp_adjoint_equivalence_to_all_invertible_2cell F₀ F₁ F₁').
- intro m.
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
apply subtypePath.
{ intro ; apply isaprop_is_invertible_2cell. }
cbn.
rewrite !vassocr.
rewrite rinvunitor_runitor, id2_left.
rewrite !vassocl.
rewrite linvunitor_lunitor, id2_right.
reflexivity.
- intro m.
use subtypePath.
{
intro.
apply isaprop_disp_left_adjoint_equivalence.
+ exact (ps_base_is_univalent_2_1 _ _ HD_2_1).
+ apply map1cells_disp_univalent_2_1.
}
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
apply subtypePath.
{ intro ; apply isaprop_is_invertible_2cell. }
cbn.
rewrite !vassocr.
rewrite lunitor_linvunitor, id2_left.
rewrite !vassocl.
rewrite runitor_rinvunitor, id2_right.
reflexivity.
Defined.
Definition map1cells_disp_univalent_2_0
(HD_2_1 : is_univalent_2_1 D)
: disp_univalent_2_0 map1cells_disp_bicat.
Proof.
apply fiberwise_univalent_2_0_to_disp_univalent_2_0.
intros F₀ F₁ F₁'.
use weqhomot.
- simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq.
simple refine (_ ∘ weqonsecfibers _ _ _)%weq.
+ exact (λ X, ∏ (Y : C) (f : X --> Y), F₁ X Y f = F₁' X Y f).
+ intro X ; cbn.
simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq.
simple refine (weqonsecfibers _ _ _)%weq.
intro Y ; cbn.
simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq.
apply idweq.
+ refine (_ ∘ weqonsecfibers _ _ _)%weq.
intro X ; cbn.
refine (weqonsecfibers _ _ _).
intro Y ; cbn.
simple refine (weqonsecfibers _ _ _).
× exact (λ f, invertible_2cell (F₁ X Y f) (F₁' X Y f)).
× intro f ; cbn.
exact (make_weq (idtoiso_2_1 (F₁ X Y f) (F₁' X Y f)) (HD_2_1 _ _ _ _)).
× exact (all_invertible_2cell_is_disp_adjoint_equivalence HD_2_1 F₀ F₁ F₁').
- intro p.
induction p.
apply subtypePath.
{
intro.
apply isaprop_disp_left_adjoint_equivalence.
+ exact (ps_base_is_univalent_2_1 _ _ HD_2_1).
+ apply map1cells_disp_univalent_2_1.
}
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
apply subtypePath.
{ intro ; apply isaprop_is_invertible_2cell. }
cbn.
rewrite id2_right.
reflexivity.
Defined.
Definition map1cells := total_bicat map1cells_disp_bicat.
Definition map1cells_is_univalent_2_1
(HD_2_1 : is_univalent_2_1 D)
: is_univalent_2_1 map1cells.
Proof.
apply total_is_univalent_2_1.
- apply ps_base_is_univalent_2_1.
exact HD_2_1.
- exact map1cells_disp_univalent_2_1.
Defined.
Definition map1cells_is_univalent_2_0
(HD : is_univalent_2 D)
: is_univalent_2_0 map1cells.
Proof.
apply total_is_univalent_2_0.
- apply ps_base_is_univalent_2. exact HD.
- exact (map1cells_disp_univalent_2_0 (pr2 HD)).
Defined.
Definition map1cells_is_univalent_2
(HD : is_univalent_2 D)
: is_univalent_2 map1cells.
Proof.
split.
- apply map1cells_is_univalent_2_0; assumption.
- apply map1cells_is_univalent_2_1.
exact (pr2 HD).
Defined.
End Map1Cells.
Definition Fobj
{C D : bicat}
(F : map1cells C D)
: C → D
:= pr1 F.
Definition Fmor
{C D : bicat}
(F : map1cells C D)
: ∏ {X Y : C}, X --> Y → Fobj F X --> Fobj F Y
:= pr2 F.
Definition ηobj
{C D : bicat}
{F G : map1cells C D}
(η : F --> G)
: ∏ (X : C), Fobj F X --> Fobj G X
:= pr1 η.
Definition ηmor
{C D : bicat}
{F G : map1cells C D}
(η : F --> G)
: ∏ {X Y : C} (f : X --> Y), ηobj η X · Fmor G f ==> Fmor F f · ηobj η Y
:= pr2 η.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Require Import UniMath.Bicategories.Core.Bicat.
Import Bicat.Notations.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat.
Import DispBicat.Notations.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Base.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.Unitors.
Require Import UniMath.Bicategories.Core.Adjunctions.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions.
Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles.
Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.
Local Open Scope cat.
Section Map1Cells.
Variable (C D : bicat).
Definition map1cells_disp_cat : disp_cat_ob_mor (ps_base C D).
Proof.
use tpair.
- exact (λ F₀, ∏ (X Y : C), X --> Y → F₀ X --> F₀ Y).
- exact (λ F₀ G₀ F₁ G₁ η, ∏ (X Y : C) (f : X --> Y),
invertible_2cell (η X · G₁ X Y f) (F₁ X Y f · η Y)).
Defined.
Definition map1cells_disp_cat_id_comp : disp_cat_id_comp (ps_base C D) map1cells_disp_cat.
Proof.
use tpair.
- cbn.
refine (λ F₀ F₁ X Y f, (lunitor (F₁ X Y f) • rinvunitor (F₁ X Y f) ,, _)).
is_iso.
- cbn.
refine (λ F₀ G₀ H₀ η₁ ε₁ F₁ G₁ H₁ η₂ ε₂ X Y f,
(rassociator (η₁ X) (ε₁ X) (H₁ X Y f))
• (η₁ X ◃ ε₂ X Y f)
• lassociator (η₁ X) (G₁ X Y f) (ε₁ Y)
• (η₂ X Y f ▹ ε₁ Y)
• rassociator (F₁ X Y f) (η₁ Y) (ε₁ Y) ,, _).
is_iso.
+ apply ε₂.
+ apply η₂.
Defined.
Definition map1cells_disp_cat_2cell : disp_2cell_struct map1cells_disp_cat
:= λ F₀ G₀ η₁ ε₁ m F₁ G₁ η₂ ε₂,
∏ (X Y : C) (f : X --> Y),
η₂ X Y f • (F₁ X Y f ◃ m Y)
=
(m X ▹ G₁ X Y f) • ε₂ X Y f.
Definition map1cells_prebicat_1 : disp_prebicat_1_id_comp_cells (ps_base C D).
Proof.
use tpair.
- use tpair.
+ exact map1cells_disp_cat.
+ exact map1cells_disp_cat_id_comp.
- exact (λ F₀ G₀ η₁ ε₁ m F₁ G₁ η₂ ε₂,
∏ (X Y : C) (f : X --> Y),
η₂ X Y f • (F₁ X Y f ◃ m Y)
=
(m X ▹ G₁ X Y f) • ε₂ X Y f).
Defined.
Definition map1cells_ops : disp_prebicat_ops map1cells_prebicat_1.
Proof.
repeat split.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite lwhisker_id2, id2_right.
rewrite id2_rwhisker, id2_left.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocl.
rewrite (lwhisker_hcomp _ (lunitor _)).
rewrite triangle_l.
rewrite <- !rwhisker_hcomp.
rewrite rwhisker_vcomp.
rewrite !vassocl.
rewrite rinvunitor_runitor, id2_right.
rewrite lunitor_triangle.
rewrite vcomp_lunitor.
use vcomp_move_R_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
rewrite <- lunitor_assoc.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocl.
use vcomp_move_R_pM.
{ is_iso. }
cbn.
refine (_ @ vassocl _ _ _).
rewrite !runitor_triangle.
rewrite (rwhisker_hcomp _ (runitor _)).
rewrite <- triangle_r.
rewrite vcomp_runitor.
rewrite <- lwhisker_vcomp, <- lwhisker_hcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite rinvunitor_triangle.
rewrite rinvunitor_runitor, id2_left.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocr.
use vcomp_move_L_Mp.
{ is_iso. }
cbn.
refine (vassocl _ _ _ @ _).
rewrite <- linvunitor_assoc.
rewrite lwhisker_hcomp.
rewrite triangle_l_inv, <- rwhisker_hcomp.
rewrite <- rwhisker_vcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite lunitor_triangle.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite vcomp_lunitor.
rewrite !vassocr.
rewrite linvunitor_lunitor, id2_left.
reflexivity.
- intros F₀ G₀ η₁ F₁ G₁ η₂ X Y f ; cbn in ×.
rewrite !vassocr.
use vcomp_move_L_Mp.
{ is_iso. }
cbn.
refine (vassocl _ _ _ @ _).
rewrite rinvunitor_triangle.
rewrite (rwhisker_hcomp _ (rinvunitor _)).
rewrite <- triangle_r_inv.
rewrite <- lwhisker_hcomp.
rewrite lwhisker_vcomp.
rewrite !vassocr.
rewrite linvunitor_lunitor, id2_left.
rewrite rinvunitor_triangle.
rewrite rinvunitor_natural.
rewrite <- rwhisker_hcomp.
reflexivity.
- intros F₀ G₀ H₀ K₀ α₁ η₁ ε₁ F₁ G₁ H₁ K₁ α₂ η₂ ε₂ X Y f ; cbn in ×.
rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp.
rewrite !vassocl.
refine (!(_ @ _)).
{
rewrite !vassocr.
do 7 apply maponpaths_2.
symmetry.
rewrite lwhisker_hcomp, rwhisker_hcomp.
rewrite vassocl.
apply inverse_pentagon.
}
rewrite !vassocl.
apply maponpaths.
rewrite !vassocr.
rewrite lwhisker_lwhisker_rassociator.
rewrite !vassocl.
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 5 apply maponpaths_2.
rewrite lwhisker_hcomp.
rewrite vassocl, <- inverse_pentagon_6.
rewrite <- rwhisker_hcomp.
reflexivity.
}
rewrite !vassocl.
apply maponpaths.
etrans.
{
rewrite !vassocr.
rewrite rwhisker_lwhisker_rassociator.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 3 apply maponpaths_2.
rewrite !vassocl.
rewrite lwhisker_hcomp, rwhisker_hcomp.
symmetry.
apply inverse_pentagon.
}
rewrite (lwhisker_hcomp _ (rassociator _ _ _)), (rwhisker_hcomp _ (rassociator _ _ _)).
rewrite <- inverse_pentagon.
rewrite !vassocr.
apply maponpaths_2.
rewrite rwhisker_rwhisker_alt.
apply maponpaths_2.
rewrite !vassocl.
rewrite rassociator_lassociator, id2_right.
reflexivity.
- intros F₀ G₀ H₀ K₀ α₁ η₁ ε₁ F₁ G₁ H₁ K₁ α₂ η₂ ε₂ X Y f ; cbn in ×.
rewrite <- !lwhisker_vcomp, <- !rwhisker_vcomp.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 8 apply maponpaths_2.
rewrite lwhisker_hcomp, rwhisker_hcomp.
symmetry.
rewrite !vassocl.
apply inverse_pentagon.
}
rewrite !vassocl.
apply maponpaths.
etrans.
{
rewrite !vassocr.
rewrite lwhisker_lwhisker_rassociator.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
rewrite inverse_pentagon.
rewrite <- lwhisker_hcomp, <- rwhisker_hcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite lwhisker_vcomp.
rewrite rassociator_lassociator.
rewrite lwhisker_id2, id2_left.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
etrans.
{
rewrite !vassocr.
rewrite rwhisker_lwhisker_rassociator.
rewrite !vassocl.
reflexivity.
}
apply maponpaths.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
etrans.
{
rewrite !vassocr.
do 4 apply maponpaths_2.
rewrite vassocl, lwhisker_hcomp, rwhisker_hcomp.
symmetry.
apply inverse_pentagon.
}
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite rassociator_lassociator, id2_left.
rewrite !vassocr.
rewrite <- rwhisker_rwhisker_alt.
rewrite !vassocl.
apply maponpaths.
symmetry.
rewrite lwhisker_hcomp, rwhisker_hcomp.
apply inverse_pentagon_6.
- intros F₀ G₀ α₁ η₁ ε₁ m₂ n₂ F₁ G₁ α₂ η₂ ε₂ m₃ n₃ X Y f ; cbn in ×.
rewrite <- lwhisker_vcomp.
rewrite !vassocr.
rewrite m₃.
rewrite !vassocl.
rewrite n₃.
rewrite !vassocr.
rewrite rwhisker_vcomp.
reflexivity.
- intros F₀ G₀ H₀ α₁ η₁ ε₁ m₂ F₁ G₁ H₁ α₂ η₂ ε₂ m₃ X Y f ; cbn in ×.
rewrite !vassocr.
rewrite <- rwhisker_lwhisker_rassociator.
rewrite !vassocl.
apply maponpaths.
rewrite lwhisker_lwhisker_rassociator.
rewrite !vassocr.
apply maponpaths_2.
rewrite lwhisker_vcomp.
rewrite <- m₃.
rewrite <- lwhisker_vcomp.
rewrite !vassocl.
apply maponpaths.
rewrite vcomp_whisker.
rewrite !vassocr.
rewrite lwhisker_lwhisker.
reflexivity.
- intros F₀ G₀ H₀ α₁ η₁ ε₁ m₂ F₁ G₁ H₁ α₂ η₂ ε₂ m₃ X Y f ; cbn in ×.
rewrite !vassocr.
rewrite rwhisker_rwhisker_alt.
rewrite !vassocl.
apply maponpaths.
rewrite rwhisker_lwhisker_rassociator.
rewrite !vassocr.
apply maponpaths_2.
rewrite vcomp_whisker.
rewrite !vassocl.
apply maponpaths.
rewrite rwhisker_vcomp.
rewrite m₃.
rewrite <- rwhisker_vcomp.
rewrite !vassocr.
rewrite <- rwhisker_rwhisker.
reflexivity.
Qed.
Definition map1cells_ops_laws : disp_prebicat_laws (_ ,, map1cells_ops).
Proof.
repeat split ; intro ; intros ; do 3 (apply funextsec ; intro) ; apply D.
Qed.
Definition map1cells_disp_prebicat : disp_prebicat (ps_base C D)
:= (_ ,, map1cells_ops_laws).
Definition map1cells_disp_bicat : disp_bicat (ps_base C D).
Proof.
refine (map1cells_disp_prebicat ,, _).
intros X Y f g α hX hY hf hg hα hβ.
apply isasetaprop.
do 3 (apply impred ; intro).
apply D.
Defined.
Definition map1cells_disp_univalent_2_1
: disp_univalent_2_1 map1cells_disp_bicat.
Proof.
apply fiberwise_local_univalent_is_univalent_2_1.
intros F G η F₁ G₁ η₁ η₁'.
use isweqimplimpl.
- intro m ; cbn in ×.
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
pose (pr1 m X Y f) as n.
cbn in n.
rewrite id2_rwhisker, lwhisker_id2 in n.
rewrite id2_left, id2_right in n.
apply subtypePath.
+ intro.
apply isaprop_is_invertible_2cell.
+ apply n.
- repeat (apply impred_isaset ; intro).
use isaset_total2.
+ apply D.
+ intro.
apply isasetaprop.
apply isaprop_is_invertible_2cell.
- apply isaproptotal2.
+ intro.
apply (@isaprop_is_disp_invertible_2cell (ps_base C D)).
+ intros.
repeat (apply funextsec ; intro).
apply D.
Defined.
Definition all_invertible_map1cells_inv
{F G : ps_base C D}
{η ε : F --> G}
(m : invertible_2cell η ε)
{F₁ : map1cells_disp_bicat F}
{G₁ : map1cells_disp_bicat G}
{η₁ : F₁ -->[ η ] G₁}
{ε₁ : F₁ -->[ ε ] G₁}
(m₁ : η₁ ==>[ m ] ε₁)
: ε₁ ==>[ m^-1 ] η₁.
Proof.
intros X Y f.
use vcomp_move_R_Mp.
{
is_iso.
apply is_invertible_2cell_to_all_is_invertible.
is_iso.
}
rewrite !vassocl.
use vcomp_move_L_pM.
{
is_iso.
apply is_invertible_2cell_to_all_is_invertible.
is_iso.
}
exact (!(m₁ X Y f)).
Qed.
Definition all_invertible_map1cells
{F G : ps_base C D}
{η ε : F --> G}
(m : invertible_2cell η ε)
{F₁ : map1cells_disp_bicat F}
{G₁ : map1cells_disp_bicat G}
{η₁ : F₁ -->[ η ] G₁}
{ε₁ : F₁ -->[ ε ] G₁}
(m₁ : η₁ ==>[ m ] ε₁)
: is_disp_invertible_2cell m m₁.
Proof.
use tpair.
- exact (all_invertible_map1cells_inv m m₁).
- split ; repeat (apply funextsec ; intro) ; apply D.
Qed.
Section AllInvertible2CellToDispAdjEquiv.
Variable (F₀ : ps_base C D)
(F₁ F₁' : map1cells_disp_bicat F₀)
(η : (∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f))).
Local Definition all_invertible_left_adj
: F₁ -->[ internal_adjoint_equivalence_identity F₀] F₁'.
Proof.
intros X Y f ; cbn.
use tpair.
- exact (lunitor _ • (η X Y f)^-1 • rinvunitor _).
- cbn.
is_iso.
Defined.
Local Definition all_invertible_right_adj
: F₁' -->[ left_adjoint_right_adjoint (internal_adjoint_equivalence_identity F₀)] F₁.
Proof.
intros X Y f.
use tpair.
- exact (lunitor _ • η X Y f • rinvunitor _).
- cbn.
is_iso.
apply η.
Defined.
Local Definition all_invertible_unit
: id_disp F₁ ==>[ left_adjoint_unit (internal_adjoint_equivalence_identity F₀)]
all_invertible_left_adj;;all_invertible_right_adj.
Proof.
intros X Y f ; cbn.
rewrite !vassocr.
rewrite <- linvunitor_assoc.
rewrite !lwhisker_hcomp.
rewrite <- linvunitor_natural.
rewrite !vassocl.
apply maponpaths.
rewrite vassocr.
rewrite rinvunitor_natural.
rewrite !vassocl.
apply maponpaths.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite linvunitor_assoc.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite rassociator_lassociator, id2_left.
rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)).
rewrite rwhisker_vcomp.
rewrite !vassocr.
rewrite linvunitor_lunitor, id2_left.
rewrite <- rwhisker_hcomp, rwhisker_vcomp.
rewrite !vassocr.
rewrite vcomp_rinv, id2_left.
rewrite triangle_r_inv.
rewrite rwhisker_hcomp.
reflexivity.
Qed.
Local Definition all_invertible_counit
: (all_invertible_right_adj;; all_invertible_left_adj)
==>[left_adjoint_counit (internal_adjoint_equivalence_identity F₀)] id_disp F₁'.
Proof.
intros X Y f ; cbn.
rewrite <- !lwhisker_vcomp.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • (_ • z))) (vassocr _ _ _)).
rewrite rinvunitor_triangle.
rewrite rwhisker_hcomp.
rewrite <- rinvunitor_natural.
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • (_ • z))) (vassocr _ _ _)).
rewrite <- (vcomp_lunitor (F₁ X Y f)).
rewrite !vassocl.
rewrite !(maponpaths (λ z, _ • (_ • z)) (vassocr _ _ _)).
rewrite lwhisker_vcomp.
rewrite vcomp_linv, lwhisker_id2, id2_left.
rewrite !vassocl.
rewrite lunitor_runitor_identity.
rewrite runitor_triangle.
rewrite rinvunitor_runitor, id2_right.
rewrite !vassocr.
do 2 (apply maponpaths_2).
use vcomp_move_R_pM.
{ is_iso. }
cbn.
rewrite lwhisker_hcomp, rwhisker_hcomp.
rewrite triangle_r.
rewrite lunitor_runitor_identity.
reflexivity.
Qed.
Definition all_invertible_2cell_to_disp_adjoint_equivalence
: disp_adjoint_equivalence (internal_adjoint_equivalence_identity F₀) F₁ F₁'.
Proof.
use tpair.
- exact all_invertible_left_adj.
- use tpair.
+ use tpair.
× exact all_invertible_right_adj.
× split.
** exact all_invertible_unit.
** exact all_invertible_counit.
+ split ; split.
× repeat (apply funextsec ; intro).
apply D.
× repeat (apply funextsec ; intro).
apply D.
× apply all_invertible_map1cells.
× apply all_invertible_map1cells.
Defined.
End AllInvertible2CellToDispAdjEquiv.
Definition disp_adjoint_equivalence_to_all_invertible_2cell
(F₀ : ps_base C D)
(F₁ F₁' : map1cells_disp_bicat F₀)
: disp_adjoint_equivalence (internal_adjoint_equivalence_identity F₀) F₁ F₁'
→
(∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f)).
Proof.
intros m X Y f.
use tpair.
- refine (rinvunitor _ • _ • lunitor _).
exact ((pr1 m X Y f)^-1).
- cbn ; is_iso.
Defined.
Definition all_invertible_2cell_is_disp_adjoint_equivalence
(HD_2_1 : is_univalent_2_1 D)
(F₀ : ps_base C D)
(F₁ F₁' : map1cells_disp_bicat F₀)
: (∏ (X Y : C) (f : X --> Y), invertible_2cell (F₁ X Y f) (F₁' X Y f))
≃
disp_adjoint_equivalence (internal_adjoint_equivalence_identity F₀) F₁ F₁'.
Proof.
refine (make_weq (all_invertible_2cell_to_disp_adjoint_equivalence F₀ F₁ F₁') _).
use isweq_iso.
- exact (disp_adjoint_equivalence_to_all_invertible_2cell F₀ F₁ F₁').
- intro m.
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
apply subtypePath.
{ intro ; apply isaprop_is_invertible_2cell. }
cbn.
rewrite !vassocr.
rewrite rinvunitor_runitor, id2_left.
rewrite !vassocl.
rewrite linvunitor_lunitor, id2_right.
reflexivity.
- intro m.
use subtypePath.
{
intro.
apply isaprop_disp_left_adjoint_equivalence.
+ exact (ps_base_is_univalent_2_1 _ _ HD_2_1).
+ apply map1cells_disp_univalent_2_1.
}
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
apply subtypePath.
{ intro ; apply isaprop_is_invertible_2cell. }
cbn.
rewrite !vassocr.
rewrite lunitor_linvunitor, id2_left.
rewrite !vassocl.
rewrite runitor_rinvunitor, id2_right.
reflexivity.
Defined.
Definition map1cells_disp_univalent_2_0
(HD_2_1 : is_univalent_2_1 D)
: disp_univalent_2_0 map1cells_disp_bicat.
Proof.
apply fiberwise_univalent_2_0_to_disp_univalent_2_0.
intros F₀ F₁ F₁'.
use weqhomot.
- simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq.
simple refine (_ ∘ weqonsecfibers _ _ _)%weq.
+ exact (λ X, ∏ (Y : C) (f : X --> Y), F₁ X Y f = F₁' X Y f).
+ intro X ; cbn.
simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq.
simple refine (weqonsecfibers _ _ _)%weq.
intro Y ; cbn.
simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq.
apply idweq.
+ refine (_ ∘ weqonsecfibers _ _ _)%weq.
intro X ; cbn.
refine (weqonsecfibers _ _ _).
intro Y ; cbn.
simple refine (weqonsecfibers _ _ _).
× exact (λ f, invertible_2cell (F₁ X Y f) (F₁' X Y f)).
× intro f ; cbn.
exact (make_weq (idtoiso_2_1 (F₁ X Y f) (F₁' X Y f)) (HD_2_1 _ _ _ _)).
× exact (all_invertible_2cell_is_disp_adjoint_equivalence HD_2_1 F₀ F₁ F₁').
- intro p.
induction p.
apply subtypePath.
{
intro.
apply isaprop_disp_left_adjoint_equivalence.
+ exact (ps_base_is_univalent_2_1 _ _ HD_2_1).
+ apply map1cells_disp_univalent_2_1.
}
apply funextsec ; intro X.
apply funextsec ; intro Y.
apply funextsec ; intro f.
apply subtypePath.
{ intro ; apply isaprop_is_invertible_2cell. }
cbn.
rewrite id2_right.
reflexivity.
Defined.
Definition map1cells := total_bicat map1cells_disp_bicat.
Definition map1cells_is_univalent_2_1
(HD_2_1 : is_univalent_2_1 D)
: is_univalent_2_1 map1cells.
Proof.
apply total_is_univalent_2_1.
- apply ps_base_is_univalent_2_1.
exact HD_2_1.
- exact map1cells_disp_univalent_2_1.
Defined.
Definition map1cells_is_univalent_2_0
(HD : is_univalent_2 D)
: is_univalent_2_0 map1cells.
Proof.
apply total_is_univalent_2_0.
- apply ps_base_is_univalent_2. exact HD.
- exact (map1cells_disp_univalent_2_0 (pr2 HD)).
Defined.
Definition map1cells_is_univalent_2
(HD : is_univalent_2 D)
: is_univalent_2 map1cells.
Proof.
split.
- apply map1cells_is_univalent_2_0; assumption.
- apply map1cells_is_univalent_2_1.
exact (pr2 HD).
Defined.
End Map1Cells.
Definition Fobj
{C D : bicat}
(F : map1cells C D)
: C → D
:= pr1 F.
Definition Fmor
{C D : bicat}
(F : map1cells C D)
: ∏ {X Y : C}, X --> Y → Fobj F X --> Fobj F Y
:= pr2 F.
Definition ηobj
{C D : bicat}
{F G : map1cells C D}
(η : F --> G)
: ∏ (X : C), Fobj F X --> Fobj G X
:= pr1 η.
Definition ηmor
{C D : bicat}
{F G : map1cells C D}
(η : F --> G)
: ∏ {X Y : C} (f : X --> Y), ηobj η X · Fmor G f ==> Fmor F f · ηobj η Y
:= pr2 η.