Library UniMath.Bicategories.PseudoFunctors.PseudoFunctor
Pseudofunctors on bicategories
Benedikt Ahrens, Marco Maggesi February 2018Pseudo functors.
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
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.Examples.BicatOfUnivCats.
Require Import UniMath.Bicategories.Core.Invertible_2cells.
Require Import UniMath.Bicategories.Core.Univalence.
Require Import UniMath.Bicategories.Core.BicategoryLaws.
Require Import UniMath.Bicategories.Core.Adjunctions.
Require Import UniMath.Bicategories.DisplayedBicats.DispBicat.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Base.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map1Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Map2Cells.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Identitor.
Require Import UniMath.Bicategories.PseudoFunctors.Display.Compositor.
Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat.
Local Open Scope bicategory_scope.
Local Open Scope cat.
Definition psfunctor
(C D : bicat)
: UU
:= psfunctor_bicat C D.
Definition make_psfunctor_data
{C D : bicat}
(F₀ : C → D)
(F₁ : ∏ {a b : C}, C⟦a,b⟧ → D⟦F₀ a, F₀ b⟧)
(F₂ : ∏ {a b : C} {f g : C⟦a,b⟧}, f ==> g → F₁ f ==> F₁ g)
(Fid : ∏ (a : C), identity (F₀ a) ==> F₁ (identity a))
(Fcomp : (∏ (a b c : C) (f : a --> b) (g : b --> c),
F₁ f · F₁ g ==> F₁ (f · g)))
: psfunctor_data C D.
Proof.
exact ((F₀ ,, F₁) ,, (F₂ ,, Fid ,, Fcomp)).
Defined.
Definition make_psfunctor
{C D : bicat}
(F : psfunctor_data C D)
(HF : psfunctor_laws F)
(Fcells : invertible_cells F)
: psfunctor C D
:= (F ,, (HF ,, Fcells)).
Coercion psfunctor_to_psfunctor_data
{C D : bicat}
(F : psfunctor C D)
: psfunctor_data C D
:= pr1 F.
Definition psfunctor_on_cells
{C D : bicat}
(F : psfunctor C D)
{a b : C}
{f g : a --> b}
(x : f ==> g)
: #F f ==> #F g
:= pr12 (pr1 F) a b f g x.
Definition psfunctor_id
{C D : bicat}
(F : psfunctor C D)
(a : C)
: invertible_2cell (identity (F a)) (#F (identity a)).
Proof.
refine (pr122 (pr1 F) a ,, _).
apply F.
Defined.
Definition psfunctor_comp
{C D : bicat}
(F : psfunctor C D)
{a b c : C}
(f : a --> b)
(g : b --> c)
: invertible_2cell (#F f · #F g) (#F (f · g)).
Proof.
refine (pr222 (pr1 F) a b c f g ,, _).
apply F.
Defined.
Local Notation "'##'" := (psfunctor_on_cells).
Section Projection.
Context {C D : bicat}.
Variable (F : psfunctor C D).
Definition psfunctor_id2
: ∏ {a b : C} (f : a --> b), ##F (id2 f) = id2 (#F f)
:= pr1(pr12 F).
Definition psfunctor_vcomp
: ∏ {a b : C} {f g h : C⟦a, b⟧} (η : f ==> g) (φ : g ==> h),
##F (η • φ) = ##F η • ##F φ
:= pr12(pr12 F).
Definition psfunctor_lunitor
: ∏ {a b : C} (f : C⟦a, b⟧),
lunitor (#F f)
=
(psfunctor_id F a ▹ #F f)
• psfunctor_comp F (identity a) f
• ##F (lunitor f)
:= pr122(pr12 F).
Definition psfunctor_runitor
: ∏ {a b : C} (f : C⟦a, b⟧),
runitor (#F f)
=
(#F f ◃ psfunctor_id F b)
• psfunctor_comp F f (identity b)
• ##F (runitor f)
:= pr1(pr222(pr12 F)).
Definition psfunctor_lassociator
: ∏ {a b c d : C} (f : C⟦a, b⟧) (g : C⟦b, c⟧) (h : C⟦c, d⟧) ,
(#F f ◃ psfunctor_comp F g h)
• psfunctor_comp F f (g · h)
• ##F (lassociator f g h)
=
(lassociator (#F f) (#F g) (#F h))
• (psfunctor_comp F f g ▹ #F h)
• psfunctor_comp F (f · g) h
:= pr12(pr222(pr12 F)).
Definition psfunctor_lwhisker
: ∏ {a b c : C} (f : C⟦a, b⟧) {g₁ g₂ : C⟦b, c⟧} (η : g₁ ==> g₂),
psfunctor_comp F f g₁ • ##F (f ◃ η)
=
#F f ◃ ##F η • psfunctor_comp F f g₂
:= pr122(pr222(pr12 F)).
Definition psfunctor_rwhisker
: ∏ {a b c : C} {f₁ f₂ : C⟦a, b⟧} (g : C⟦b, c⟧) (η : f₁ ==> f₂),
psfunctor_comp F f₁ g • ##F (η ▹ g)
=
##F η ▹ #F g • psfunctor_comp F f₂ g
:= pr222(pr222(pr12 F)).
End Projection.
Isos are preserved
Definition psfunctor_is_iso
{C D : bicat}
(F : psfunctor C D)
{a b : C}
{f g : C⟦a,b⟧}
(α : invertible_2cell f g)
: is_invertible_2cell (##F α).
Proof.
use tpair.
- exact (##F (α^-1)).
- split ; cbn
; rewrite <- psfunctor_vcomp, <- psfunctor_id2 ; apply maponpaths.
+ apply vcomp_rinv.
+ apply vcomp_linv.
Defined.
Section PseudoFunctorDerivedLaws.
Context {C D : bicat}.
Variable (F : psfunctor C D).
Definition psfunctor_linvunitor
{a b : C}
(f : C⟦a, b⟧)
: ##F (linvunitor f)
=
(linvunitor (#F f))
• ((psfunctor_id F a) ▹ #F f)
• (psfunctor_comp F _ _).
Proof.
rewrite !vassocl.
cbn.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
use vcomp_move_R_Mp.
{
refine (psfunctor_is_iso F (linvunitor f ,, _)).
is_iso.
}
cbn.
rewrite psfunctor_lunitor ; cbn.
rewrite <- !vassocr.
reflexivity.
Qed.
Definition psfunctor_rinvunitor
{a b : C}
(f : C⟦a, b⟧)
: ##F (rinvunitor f)
=
(rinvunitor (#F f))
• (#F f ◃ psfunctor_id F b)
• psfunctor_comp F _ _.
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
use vcomp_move_R_Mp.
{
refine (psfunctor_is_iso F (rinvunitor f ,, _)).
is_iso.
}
cbn.
rewrite psfunctor_runitor ; cbn.
rewrite <- !vassocr.
reflexivity.
Qed.
Definition psfunctor_rassociator
{a b c d : C}
(f : C⟦a, b⟧) (g : C⟦b, c⟧) (h : C⟦c, d⟧)
: (psfunctor_comp F f g ▹ #F h)
• psfunctor_comp F (f · g) h
• ##F (rassociator f g h)
=
(rassociator (#F f) (#F g) (#F h))
• (#F f ◃ psfunctor_comp F g h)
• psfunctor_comp F f (g · h).
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
use vcomp_move_R_Mp.
{ refine (psfunctor_is_iso F (rassociator f g h ,, _)).
is_iso.
}
cbn.
symmetry.
exact (psfunctor_lassociator F f g h).
Qed.
Definition psfunctor_comp_natural
{a b c : C}
{g₁ g₂ : C⟦b,c⟧} {f₁ f₂ : C⟦a,b⟧}
(ηg : g₁ ==> g₂) (ηf : f₁ ==> f₂)
: psfunctor_comp F f₁ g₁ • ##F (ηf ⋆ ηg)
=
(##F ηf) ⋆ (##F ηg) • psfunctor_comp F f₂ g₂.
Proof.
unfold hcomp.
rewrite !psfunctor_vcomp.
rewrite !vassocr.
rewrite !psfunctor_rwhisker.
rewrite !vassocl.
rewrite psfunctor_lwhisker.
reflexivity.
Qed.
Definition psfunctor_F_lunitor
{a b : C}
(f : C⟦a, b⟧)
: ##F (lunitor f)
=
((psfunctor_comp F (identity a) f)^-1)
• ((psfunctor_id F a)^-1 ▹ #F f)
• lunitor (#F f).
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
exact (!(psfunctor_lunitor F f)).
Qed.
Definition psfunctor_F_runitor
{a b : C}
(f : C⟦a,b⟧)
: ##F (runitor f)
=
((psfunctor_comp F f (identity b))^-1)
• (#F f ◃ (psfunctor_id F b)^-1)
• runitor (#F f).
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
exact (!(psfunctor_runitor F f)).
Qed.
Definition psfunctor_lassociator_alt
{a b c d : C}
(f : C⟦a, b⟧) (g : C⟦b, c⟧) (h : C⟦c, d⟧)
: (psfunctor_comp F f (g · h))
• ##F (lassociator _ _ _)
• (psfunctor_comp F (f · g) h)^-1
• ((psfunctor_comp F f g)^-1 ▹ #F h)
• rassociator _ _ _
=
#F f ◃ (psfunctor_comp F g h)^-1.
Proof.
use vcomp_move_R_Mp.
{ is_iso. }
use vcomp_move_R_Mp.
{ is_iso. }
use vcomp_move_R_Mp.
{ is_iso. }
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
apply (psfunctor_lassociator F f g h).
Qed.
End PseudoFunctorDerivedLaws.
Section PseudoFunctorLocalFunctor.
Context {B C : bicat}.
Variable (F : psfunctor B C)
(X Y : B).
Definition Fmor_data
: functor_data (hom X Y) (hom (F X) (F Y)).
Proof.
use make_functor_data.
- exact (λ f, #F f).
- exact (λ _ _ α, ##F α).
Defined.
Definition Fmor_is_functor
: is_functor Fmor_data.
Proof.
split.
- intros f.
exact (psfunctor_id2 F f).
- intros f g h α β.
exact (psfunctor_vcomp F α β).
Defined.
Definition Fmor
: hom X Y ⟶ hom (F X) (F Y).
Proof.
use make_functor.
- exact Fmor_data.
- exact Fmor_is_functor.
Defined.
Definition Fmor_univ
(B_is_univalent_2_1 : is_univalent_2_1 B)
(C_is_univalent_2_1 : is_univalent_2_1 C)
: (univ_hom B_is_univalent_2_1 X Y)
⟶
univ_hom C_is_univalent_2_1 (F X) (F Y).
Proof.
exact Fmor.
Defined.
End PseudoFunctorLocalFunctor.
Section ExtendPseudoFunctor.
Context {C D : bicat}
(HD : is_univalent_2 D)
(F : psfunctor C D)
(G : ob C → ob D)
(η : ∏ (a : C), adjoint_equivalence (F a) (G a)).
Definition extend_psfunctor : psfunctor C D.
Proof.
cbn.
pose (adjequiv_base_adjequiv_tot (ps_base_is_univalent_2_0 C D HD) (adjequiv_ps_base C D F G η) (pr211 F)) as m.
induction m as [m1 m2].
pose (adjequiv_base_adjequiv_tot (map1cells_is_univalent_2_0 C D HD) m2 (pr21 F)) as n.
induction n as [n1 n2].
pose (adjequiv_base_adjequiv_tot (psfunctor_data_is_univalent_2_0 C D HD) n2 (pr2 F)) as p.
exact (_ ,, pr1 p).
Defined.
End ExtendPseudoFunctor.
Definition psfunctor_preserves_adjequiv
{C D : bicat}
(HC : is_univalent_2_0 C)
(HD : is_univalent_2_1 D)
(F : psfunctor C D)
(a b : C)
(f : adjoint_equivalence a b)
: left_adjoint_equivalence (#F f)
:= J_2_0 HC
(λ a b f, left_adjoint_equivalence (#F (pr1 f)))
(λ a0,
left_adjequiv_invertible_2cell HD _ _
(psfunctor_id F a0)
(pr2 (internal_adjoint_equivalence_identity (F a0))))
f.
Definition local_equivalence
{B₁ B₂ : bicat}
(B₁_is_univalent_2_1 : is_univalent_2_1 B₁)
(B₂_is_univalent_2_1 : is_univalent_2_1 B₂)
(F : psfunctor B₁ B₂)
: UU
:= ∏ (x y : B₁),
@left_adjoint_equivalence
bicat_of_univ_cats
_ _
(Fmor_univ
F x y
B₁_is_univalent_2_1
B₂_is_univalent_2_1).
Definition essentially_surjective
{B₁ B₂ : bicat}
(F : psfunctor B₁ B₂)
: hProp
:= ∀ (y : B₂), ∃ (x : B₁), adjoint_equivalence (F x) y.
Definition weak_equivalence
{B₁ B₂ : bicat}
(B₁_is_univalent_2_1 : is_univalent_2_1 B₁)
(B₂_is_univalent_2_1 : is_univalent_2_1 B₂)
(F : psfunctor B₁ B₂)
: UU
:= local_equivalence
B₁_is_univalent_2_1
B₂_is_univalent_2_1
F
× essentially_surjective F.
Module Notations.
Notation "'##'" := (psfunctor_on_cells).
End Notations.
{C D : bicat}
(F : psfunctor C D)
{a b : C}
{f g : C⟦a,b⟧}
(α : invertible_2cell f g)
: is_invertible_2cell (##F α).
Proof.
use tpair.
- exact (##F (α^-1)).
- split ; cbn
; rewrite <- psfunctor_vcomp, <- psfunctor_id2 ; apply maponpaths.
+ apply vcomp_rinv.
+ apply vcomp_linv.
Defined.
Section PseudoFunctorDerivedLaws.
Context {C D : bicat}.
Variable (F : psfunctor C D).
Definition psfunctor_linvunitor
{a b : C}
(f : C⟦a, b⟧)
: ##F (linvunitor f)
=
(linvunitor (#F f))
• ((psfunctor_id F a) ▹ #F f)
• (psfunctor_comp F _ _).
Proof.
rewrite !vassocl.
cbn.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
use vcomp_move_R_Mp.
{
refine (psfunctor_is_iso F (linvunitor f ,, _)).
is_iso.
}
cbn.
rewrite psfunctor_lunitor ; cbn.
rewrite <- !vassocr.
reflexivity.
Qed.
Definition psfunctor_rinvunitor
{a b : C}
(f : C⟦a, b⟧)
: ##F (rinvunitor f)
=
(rinvunitor (#F f))
• (#F f ◃ psfunctor_id F b)
• psfunctor_comp F _ _.
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
use vcomp_move_R_Mp.
{
refine (psfunctor_is_iso F (rinvunitor f ,, _)).
is_iso.
}
cbn.
rewrite psfunctor_runitor ; cbn.
rewrite <- !vassocr.
reflexivity.
Qed.
Definition psfunctor_rassociator
{a b c d : C}
(f : C⟦a, b⟧) (g : C⟦b, c⟧) (h : C⟦c, d⟧)
: (psfunctor_comp F f g ▹ #F h)
• psfunctor_comp F (f · g) h
• ##F (rassociator f g h)
=
(rassociator (#F f) (#F g) (#F h))
• (#F f ◃ psfunctor_comp F g h)
• psfunctor_comp F f (g · h).
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
use vcomp_move_R_Mp.
{ refine (psfunctor_is_iso F (rassociator f g h ,, _)).
is_iso.
}
cbn.
symmetry.
exact (psfunctor_lassociator F f g h).
Qed.
Definition psfunctor_comp_natural
{a b c : C}
{g₁ g₂ : C⟦b,c⟧} {f₁ f₂ : C⟦a,b⟧}
(ηg : g₁ ==> g₂) (ηf : f₁ ==> f₂)
: psfunctor_comp F f₁ g₁ • ##F (ηf ⋆ ηg)
=
(##F ηf) ⋆ (##F ηg) • psfunctor_comp F f₂ g₂.
Proof.
unfold hcomp.
rewrite !psfunctor_vcomp.
rewrite !vassocr.
rewrite !psfunctor_rwhisker.
rewrite !vassocl.
rewrite psfunctor_lwhisker.
reflexivity.
Qed.
Definition psfunctor_F_lunitor
{a b : C}
(f : C⟦a, b⟧)
: ##F (lunitor f)
=
((psfunctor_comp F (identity a) f)^-1)
• ((psfunctor_id F a)^-1 ▹ #F f)
• lunitor (#F f).
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
exact (!(psfunctor_lunitor F f)).
Qed.
Definition psfunctor_F_runitor
{a b : C}
(f : C⟦a,b⟧)
: ##F (runitor f)
=
((psfunctor_comp F f (identity b))^-1)
• (#F f ◃ (psfunctor_id F b)^-1)
• runitor (#F f).
Proof.
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
exact (!(psfunctor_runitor F f)).
Qed.
Definition psfunctor_lassociator_alt
{a b c d : C}
(f : C⟦a, b⟧) (g : C⟦b, c⟧) (h : C⟦c, d⟧)
: (psfunctor_comp F f (g · h))
• ##F (lassociator _ _ _)
• (psfunctor_comp F (f · g) h)^-1
• ((psfunctor_comp F f g)^-1 ▹ #F h)
• rassociator _ _ _
=
#F f ◃ (psfunctor_comp F g h)^-1.
Proof.
use vcomp_move_R_Mp.
{ is_iso. }
use vcomp_move_R_Mp.
{ is_iso. }
use vcomp_move_R_Mp.
{ is_iso. }
rewrite !vassocl.
use vcomp_move_L_pM.
{ is_iso. }
cbn.
rewrite !vassocr.
apply (psfunctor_lassociator F f g h).
Qed.
End PseudoFunctorDerivedLaws.
Section PseudoFunctorLocalFunctor.
Context {B C : bicat}.
Variable (F : psfunctor B C)
(X Y : B).
Definition Fmor_data
: functor_data (hom X Y) (hom (F X) (F Y)).
Proof.
use make_functor_data.
- exact (λ f, #F f).
- exact (λ _ _ α, ##F α).
Defined.
Definition Fmor_is_functor
: is_functor Fmor_data.
Proof.
split.
- intros f.
exact (psfunctor_id2 F f).
- intros f g h α β.
exact (psfunctor_vcomp F α β).
Defined.
Definition Fmor
: hom X Y ⟶ hom (F X) (F Y).
Proof.
use make_functor.
- exact Fmor_data.
- exact Fmor_is_functor.
Defined.
Definition Fmor_univ
(B_is_univalent_2_1 : is_univalent_2_1 B)
(C_is_univalent_2_1 : is_univalent_2_1 C)
: (univ_hom B_is_univalent_2_1 X Y)
⟶
univ_hom C_is_univalent_2_1 (F X) (F Y).
Proof.
exact Fmor.
Defined.
End PseudoFunctorLocalFunctor.
Section ExtendPseudoFunctor.
Context {C D : bicat}
(HD : is_univalent_2 D)
(F : psfunctor C D)
(G : ob C → ob D)
(η : ∏ (a : C), adjoint_equivalence (F a) (G a)).
Definition extend_psfunctor : psfunctor C D.
Proof.
cbn.
pose (adjequiv_base_adjequiv_tot (ps_base_is_univalent_2_0 C D HD) (adjequiv_ps_base C D F G η) (pr211 F)) as m.
induction m as [m1 m2].
pose (adjequiv_base_adjequiv_tot (map1cells_is_univalent_2_0 C D HD) m2 (pr21 F)) as n.
induction n as [n1 n2].
pose (adjequiv_base_adjequiv_tot (psfunctor_data_is_univalent_2_0 C D HD) n2 (pr2 F)) as p.
exact (_ ,, pr1 p).
Defined.
End ExtendPseudoFunctor.
Definition psfunctor_preserves_adjequiv
{C D : bicat}
(HC : is_univalent_2_0 C)
(HD : is_univalent_2_1 D)
(F : psfunctor C D)
(a b : C)
(f : adjoint_equivalence a b)
: left_adjoint_equivalence (#F f)
:= J_2_0 HC
(λ a b f, left_adjoint_equivalence (#F (pr1 f)))
(λ a0,
left_adjequiv_invertible_2cell HD _ _
(psfunctor_id F a0)
(pr2 (internal_adjoint_equivalence_identity (F a0))))
f.
Definition local_equivalence
{B₁ B₂ : bicat}
(B₁_is_univalent_2_1 : is_univalent_2_1 B₁)
(B₂_is_univalent_2_1 : is_univalent_2_1 B₂)
(F : psfunctor B₁ B₂)
: UU
:= ∏ (x y : B₁),
@left_adjoint_equivalence
bicat_of_univ_cats
_ _
(Fmor_univ
F x y
B₁_is_univalent_2_1
B₂_is_univalent_2_1).
Definition essentially_surjective
{B₁ B₂ : bicat}
(F : psfunctor B₁ B₂)
: hProp
:= ∀ (y : B₂), ∃ (x : B₁), adjoint_equivalence (F x) y.
Definition weak_equivalence
{B₁ B₂ : bicat}
(B₁_is_univalent_2_1 : is_univalent_2_1 B₁)
(B₂_is_univalent_2_1 : is_univalent_2_1 B₂)
(F : psfunctor B₁ B₂)
: UU
:= local_equivalence
B₁_is_univalent_2_1
B₂_is_univalent_2_1
F
× essentially_surjective F.
Module Notations.
Notation "'##'" := (psfunctor_on_cells).
End Notations.