Library UniMath.CategoryTheory.Equivalences.CompositesAndInverses
Composition and inverses of (adjoint) equivalences of precategories
Contents
- Preliminaries
- Composition
- Inverses
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.MoreFoundations.PartA.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Adjunctions.Core.
Require Import UniMath.CategoryTheory.Equivalences.Core.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.CategoryTheory.PrecategoryBinProduct.
Local Open Scope cat.
Local Open Scope cat_deprecated.
Lemma is_iso_comp_is_iso {C : category} {a b c : ob C}
(f : C⟦a, b⟧) (g : C⟦b, c⟧)
: is_iso f → is_iso g → is_iso (f ;; g).
Proof.
intros Hf Hg.
apply (is_iso_comp_of_isos (make_iso f Hf) (make_iso g Hg)).
Defined.
Lemma functor_is_iso_is_iso {C C' : category} (F : functor C C')
{a b : ob C} (f : C ⟦a,b⟧) (fH : is_iso f) : is_iso (#F f).
Proof.
apply (functor_on_iso_is_iso _ _ F _ _ (make_iso f fH)).
Defined.
Coercion left_adj_from_adj_equiv (X Y : category) (K : functor X Y)
(HK : adj_equivalence_of_cats K) : is_left_adjoint K := pr1 HK.
Section A.
Variables D1 D2 : category.
Variable F : functor D1 D2.
Variable GG : adj_equivalence_of_cats F.
Let G : functor D2 D1 := right_adjoint GG.
Let η := unit_from_left_adjoint GG.
Let ε := counit_from_left_adjoint GG.
Let ηinv a := iso_inv_from_iso (unit_pointwise_iso_from_adj_equivalence GG a).
Let εinv a := iso_inv_from_iso (counit_pointwise_iso_from_adj_equivalence GG a).
Lemma right_adj_equiv_is_ff : fully_faithful G.
Proof.
intros c d.
set (inv := (fun f : D1 ⟦G c, G d⟧ ⇒ εinv _ ;; #F f ;; ε _ )).
simpl in inv.
apply (gradth _ inv ).
- intro f. simpl in f. unfold inv.
assert (XR := nat_trans_ax ε). simpl in XR.
rewrite <- assoc.
etrans. apply maponpaths. apply XR.
rewrite assoc.
etrans. apply maponpaths_2. apply iso_after_iso_inv.
apply id_left.
- intro g.
unfold inv.
do 2 rewrite functor_comp.
intermediate_path ((# G (inv_from_iso (counit_pointwise_iso_from_adj_equivalence GG c)) ;; ηinv _ ) ;; (η _ ;; # G (# F g)) ;; # G (ε d)).
+ do 4 rewrite <- assoc.
apply maponpaths.
do 2 rewrite assoc.
etrans.
2: do 2 apply maponpaths_2; eapply pathsinv0, iso_after_iso_inv.
refine (_ @ assoc _ _ _).
exact (!id_left _).
+ assert (XR := nat_trans_ax η). simpl in XR. rewrite <- XR. clear XR.
do 3 rewrite <- assoc.
etrans. do 3 apply maponpaths. apply triangle_id_right_ad. rewrite id_right.
rewrite assoc.
etrans.
2: apply id_left.
apply maponpaths_2.
etrans. apply maponpaths_2. apply functor_on_inv_from_iso.
assert (XR := triangle_id_right_ad (pr2 (pr1 GG))); simpl in XR.
unfold ηinv; simpl.
pose (XRR := maponpaths pr1 (iso_inv_of_iso_comp _ _ _ _ (unit_pointwise_iso_from_adj_equivalence GG ((adj_equivalence_inv GG) c)) (functor_on_iso G (counit_pointwise_iso_from_adj_equivalence GG c)) )).
simpl in XRR.
etrans.
apply (! XRR); clear XRR.
apply pathsinv0, inv_iso_unique'.
simpl. cbn. unfold precomp_with.
rewrite id_right. apply XR.
Defined.
Lemma right_adj_equiv_is_ess_sur : essentially_surjective G.
Proof.
intro d.
apply hinhpr.
∃ (F d).
exact (ηinv d).
Defined.
End A.
Section eqv_comp.
Context {A B C : category}
{F : functor A B}
{F' : functor B C}.
Hypothesis HF : adj_equivalence_of_cats F.
Hypothesis HF' : adj_equivalence_of_cats F'.
Definition comp_adj_equivalence_of_cats
: adj_equivalence_of_cats (functor_composite F F').
Proof.
∃ (is_left_adjoint_functor_composite HF HF').
use tpair.
- intro.
apply is_iso_comp_is_iso.
+ apply (pr1 (pr2 HF)).
+ simpl.
refine (eqweqmap (maponpaths is_iso _) _).
refine (_ @ !id_right _).
exact (!id_left _).
apply functor_is_iso_is_iso, (pr1 (pr2 HF')).
- cbn. intro. apply is_iso_comp_is_iso.
+
refine (eqweqmap (maponpaths is_iso _) _).
refine (_ @ !id_left _).
refine (_ @ !id_left _).
exact (!id_right _).
apply functor_is_iso_is_iso, (pr2 (pr2 HF)).
+ apply (pr2 (pr2 HF')).
Defined.
End eqv_comp.
Section eqv_inv.
Local Definition nat_iso_to_pointwise_iso {A B : category} {F G : functor A B}
(n : nat_iso F G) (x : ob A) : iso (F x) (G x) := make_iso _ (pr2 n x).
Local Lemma nat_iso_inv_after_nat_iso {A B : category} {F G : functor A B}
(n : nat_iso F G) : ∏ x, (nat_iso_to_pointwise_iso n) x · (nat_iso_inv n) x = identity _.
Proof.
intro; apply iso_inv_after_iso.
Qed.
Context {A B : category} {F : functor A B}
(adEquivF : adj_equivalence_of_cats F).
Local Notation η := (unit_from_left_adjoint adEquivF).
Local Notation ε := (counit_from_left_adjoint adEquivF).
Local Notation G := (right_adjoint (pr1 adEquivF)).
Local Notation ηiso := (unit_nat_iso_from_adj_equivalence_of_cats adEquivF).
Local Notation εiso := (counit_nat_iso_from_adj_equivalence_of_cats adEquivF).
Lemma form_adjunction_inv :
form_adjunction _ F (nat_iso_inv εiso) (nat_iso_inv ηiso).
Proof.
split.
- intro b.
Use the right triangle identity that we already know
Transform it by precomposing with the inverse isos
apply (pre_comp_with_iso_is_inj _ _ _
(#G (nat_iso_to_pointwise_iso εiso b)));
[apply (functor_is_iso_is_iso G), iso_is_iso |].
(#G (nat_iso_to_pointwise_iso εiso b)));
[apply (functor_is_iso_is_iso G), iso_is_iso |].
Cancel the isos
unfold adjunit; unfold adjcounit.
unfold pr2, pr1.
rewrite assoc.
rewrite <- functor_comp.
unfold left_functor; unfold pr1.
rewrite nat_iso_inv_after_nat_iso.
rewrite functor_id.
rewrite id_left.
rewrite assoc.
unfold pr2, pr1.
rewrite assoc.
rewrite <- functor_comp.
unfold left_functor; unfold pr1.
rewrite nat_iso_inv_after_nat_iso.
rewrite functor_id.
rewrite id_left.
rewrite assoc.
Again, precompose with the inverse iso
apply (pre_comp_with_iso_is_inj _ _ _
((nat_iso_to_pointwise_iso ηiso (G b))));
[apply iso_is_iso; rewrite iso_inv_after_iso|].
rewrite (nat_iso_inv_after_nat_iso ηiso).
refine (!triangle_id_right_ad (pr2 (pr1 adEquivF)) _ @ _).
refine (!id_left _ @ _).
repeat rewrite assoc.
do 2 rewrite <- assoc. apply (maponpaths (fun f ⇒ f · _)).
apply (!triangle_id_right_ad (pr2 (pr1 adEquivF)) _).
-
((nat_iso_to_pointwise_iso ηiso (G b))));
[apply iso_is_iso; rewrite iso_inv_after_iso|].
rewrite (nat_iso_inv_after_nat_iso ηiso).
refine (!triangle_id_right_ad (pr2 (pr1 adEquivF)) _ @ _).
refine (!id_left _ @ _).
repeat rewrite assoc.
do 2 rewrite <- assoc. apply (maponpaths (fun f ⇒ f · _)).
apply (!triangle_id_right_ad (pr2 (pr1 adEquivF)) _).
-
Same proof, just backwards
intro a.
refine (_ @ triangle_id_left_ad (pr2 (pr1 adEquivF)) a).
apply (pre_comp_with_iso_is_inj _ _ _
((nat_iso_to_pointwise_iso εiso (F a))));
[apply iso_is_iso; rewrite iso_inv_after_iso|].
rewrite assoc.
rewrite (nat_iso_inv_after_nat_iso εiso).
rewrite id_left.
apply (pre_comp_with_iso_is_inj _ _ _ (#F (nat_iso_to_pointwise_iso ηiso a)));
[apply (functor_is_iso_is_iso F), iso_is_iso |].
unfold adjunit; unfold adjcounit.
unfold right_functor.
unfold pr2, pr1.
refine (_ @ !assoc _ _ _).
refine (!functor_comp F _ _ @ _).
rewrite (nat_iso_inv_after_nat_iso ηiso).
rewrite functor_id.
refine (!triangle_id_left_ad (pr2 (pr1 adEquivF)) _ @ _).
refine (!id_left _ @ _).
repeat rewrite assoc.
do 2 rewrite <- assoc. apply (maponpaths (fun f ⇒ f · _)).
apply (!triangle_id_left_ad (pr2 (pr1 adEquivF)) _).
Qed.
Definition is_left_adjoint_inv : is_left_adjoint G.
Proof.
use tpair.
- apply F.
- use tpair.
∃ (pr1 (nat_iso_inv εiso)).
exact (pr1 (nat_iso_inv ηiso)).
apply form_adjunction_inv.
Defined.
Definition adj_equivalence_of_cats_inv
: adj_equivalence_of_cats G.
Proof.
∃ is_left_adjoint_inv.
split; intro; apply is_iso_inv_from_iso.
Defined.
End eqv_inv.
refine (_ @ triangle_id_left_ad (pr2 (pr1 adEquivF)) a).
apply (pre_comp_with_iso_is_inj _ _ _
((nat_iso_to_pointwise_iso εiso (F a))));
[apply iso_is_iso; rewrite iso_inv_after_iso|].
rewrite assoc.
rewrite (nat_iso_inv_after_nat_iso εiso).
rewrite id_left.
apply (pre_comp_with_iso_is_inj _ _ _ (#F (nat_iso_to_pointwise_iso ηiso a)));
[apply (functor_is_iso_is_iso F), iso_is_iso |].
unfold adjunit; unfold adjcounit.
unfold right_functor.
unfold pr2, pr1.
refine (_ @ !assoc _ _ _).
refine (!functor_comp F _ _ @ _).
rewrite (nat_iso_inv_after_nat_iso ηiso).
rewrite functor_id.
refine (!triangle_id_left_ad (pr2 (pr1 adEquivF)) _ @ _).
refine (!id_left _ @ _).
repeat rewrite assoc.
do 2 rewrite <- assoc. apply (maponpaths (fun f ⇒ f · _)).
apply (!triangle_id_left_ad (pr2 (pr1 adEquivF)) _).
Qed.
Definition is_left_adjoint_inv : is_left_adjoint G.
Proof.
use tpair.
- apply F.
- use tpair.
∃ (pr1 (nat_iso_inv εiso)).
exact (pr1 (nat_iso_inv ηiso)).
apply form_adjunction_inv.
Defined.
Definition adj_equivalence_of_cats_inv
: adj_equivalence_of_cats G.
Proof.
∃ is_left_adjoint_inv.
split; intro; apply is_iso_inv_from_iso.
Defined.
End eqv_inv.
Closure under natural isomorphisms
Definition nat_iso_equivalence_of_cats
{C₁ C₂ : category}
{F G : C₁ ⟶ C₂}
(α : F ⟹ G)
(Hα : is_nat_iso α)
(HF : adj_equivalence_of_cats F)
: equivalence_of_cats C₁ C₂.
Proof.
use make_equivalence_of_cats.
- use make_adjunction_data.
+ exact G.
+ exact (right_adjoint HF).
+ exact (nat_trans_comp
_ _ _
(adjunit HF)
(post_whisker α _)).
+ exact (nat_trans_comp
_ _ _
(pre_whisker _ (nat_iso_inv (make_nat_iso _ _ α Hα)))
(adjcounit HF)).
- split.
+ intro x ; cbn.
use is_iso_comp_of_is_isos.
× apply (unit_nat_iso_from_adj_equivalence_of_cats HF).
× apply functor_is_iso_is_iso.
apply Hα.
+ intro x ; cbn.
use is_iso_comp_of_is_isos.
× apply is_iso_inv_from_iso.
× apply (counit_nat_iso_from_adj_equivalence_of_cats HF).
Defined.
Definition nat_iso_adj_equivalence_of_cats
{C₁ C₂ : category}
{F G : C₁ ⟶ C₂}
(α : F ⟹ G)
(Hα : is_nat_iso α)
(HF : adj_equivalence_of_cats F)
: adj_equivalence_of_cats G
:= adjointificiation (nat_iso_equivalence_of_cats α Hα HF).
Section PairEquivalence.
Context {C₁ C₁' C₂ C₂' : category}
{F : C₁ ⟶ C₁'}
{G : C₂ ⟶ C₂'}
(HF : adj_equivalence_of_cats F)
(HG : adj_equivalence_of_cats G).
Let L : category_binproduct C₁ C₂ ⟶ category_binproduct C₁' C₂'
:= pair_functor F G.
Let R : category_binproduct C₁' C₂' ⟶ category_binproduct C₁ C₂
:= pair_functor (right_adjoint HF) (right_adjoint HG).
Definition pair_equivalence_of_cats_unit_data
: nat_trans_data (functor_identity _) (L ∙ R)
:= λ x, adjunit HF (pr1 x) ,, adjunit HG (pr2 x).
Definition pair_equivalence_of_cats_unit_is_nat_trans
: is_nat_trans
_ _
pair_equivalence_of_cats_unit_data.
Proof.
intros x y f.
use pathsdirprod ; cbn.
- exact (nat_trans_ax (adjunit HF) _ _ (pr1 f)).
- exact (nat_trans_ax (adjunit HG) _ _ (pr2 f)).
Qed.
Definition pair_equivalence_of_cats_unit
: functor_identity (category_binproduct C₁ C₂) ⟹ L ∙ R.
Proof.
use make_nat_trans.
- exact pair_equivalence_of_cats_unit_data.
- exact pair_equivalence_of_cats_unit_is_nat_trans.
Defined.
Definition pair_equivalence_of_cats_counit_data
: nat_trans_data (R ∙ L) (functor_identity _)
:= λ x, adjcounit HF (pr1 x) ,, adjcounit HG (pr2 x).
Definition pair_equivalence_of_cats_counit_is_nat_trans
: is_nat_trans
_ _
pair_equivalence_of_cats_counit_data.
Proof.
intros x y f.
use pathsdirprod ; cbn.
- exact (nat_trans_ax (adjcounit HF) _ _ (pr1 f)).
- exact (nat_trans_ax (adjcounit HG) _ _ (pr2 f)).
Qed.
Definition pair_equivalence_of_cats_counit
: R ∙ L ⟹ functor_identity _.
Proof.
use make_nat_trans.
- exact pair_equivalence_of_cats_counit_data.
- exact pair_equivalence_of_cats_counit_is_nat_trans.
Defined.
Definition pair_equivalence_of_cats
: equivalence_of_cats
(category_binproduct C₁ C₂)
(category_binproduct C₁' C₂').
Proof.
use make_equivalence_of_cats.
- use make_adjunction_data.
+ exact L.
+ exact R.
+ exact pair_equivalence_of_cats_unit.
+ exact pair_equivalence_of_cats_counit.
- split.
+ intro x.
use is_iso_binprod_iso.
× exact (iso_is_iso
(nat_iso_pointwise_iso
(unit_nat_iso_from_adj_equivalence_of_cats HF)
(pr1 x))).
× exact (iso_is_iso
(nat_iso_pointwise_iso
(unit_nat_iso_from_adj_equivalence_of_cats HG)
(pr2 x))).
+ intro x.
use is_iso_binprod_iso.
× exact (iso_is_iso
(nat_iso_pointwise_iso
(counit_nat_iso_from_adj_equivalence_of_cats HF)
(pr1 x))).
× exact (iso_is_iso
(nat_iso_pointwise_iso
(counit_nat_iso_from_adj_equivalence_of_cats HG)
(pr2 x))).
Defined.
End PairEquivalence.
Definition pair_adj_equivalence_of_cats
{C₁ C₁' C₂ C₂' : category}
{F : C₁ ⟶ C₁'}
{G : C₂ ⟶ C₂'}
(HF : adj_equivalence_of_cats F)
(HG : adj_equivalence_of_cats G)
: adj_equivalence_of_cats (pair_functor F G)
:= adjointificiation (pair_equivalence_of_cats HF HG).
{C₁ C₂ : category}
{F G : C₁ ⟶ C₂}
(α : F ⟹ G)
(Hα : is_nat_iso α)
(HF : adj_equivalence_of_cats F)
: equivalence_of_cats C₁ C₂.
Proof.
use make_equivalence_of_cats.
- use make_adjunction_data.
+ exact G.
+ exact (right_adjoint HF).
+ exact (nat_trans_comp
_ _ _
(adjunit HF)
(post_whisker α _)).
+ exact (nat_trans_comp
_ _ _
(pre_whisker _ (nat_iso_inv (make_nat_iso _ _ α Hα)))
(adjcounit HF)).
- split.
+ intro x ; cbn.
use is_iso_comp_of_is_isos.
× apply (unit_nat_iso_from_adj_equivalence_of_cats HF).
× apply functor_is_iso_is_iso.
apply Hα.
+ intro x ; cbn.
use is_iso_comp_of_is_isos.
× apply is_iso_inv_from_iso.
× apply (counit_nat_iso_from_adj_equivalence_of_cats HF).
Defined.
Definition nat_iso_adj_equivalence_of_cats
{C₁ C₂ : category}
{F G : C₁ ⟶ C₂}
(α : F ⟹ G)
(Hα : is_nat_iso α)
(HF : adj_equivalence_of_cats F)
: adj_equivalence_of_cats G
:= adjointificiation (nat_iso_equivalence_of_cats α Hα HF).
Section PairEquivalence.
Context {C₁ C₁' C₂ C₂' : category}
{F : C₁ ⟶ C₁'}
{G : C₂ ⟶ C₂'}
(HF : adj_equivalence_of_cats F)
(HG : adj_equivalence_of_cats G).
Let L : category_binproduct C₁ C₂ ⟶ category_binproduct C₁' C₂'
:= pair_functor F G.
Let R : category_binproduct C₁' C₂' ⟶ category_binproduct C₁ C₂
:= pair_functor (right_adjoint HF) (right_adjoint HG).
Definition pair_equivalence_of_cats_unit_data
: nat_trans_data (functor_identity _) (L ∙ R)
:= λ x, adjunit HF (pr1 x) ,, adjunit HG (pr2 x).
Definition pair_equivalence_of_cats_unit_is_nat_trans
: is_nat_trans
_ _
pair_equivalence_of_cats_unit_data.
Proof.
intros x y f.
use pathsdirprod ; cbn.
- exact (nat_trans_ax (adjunit HF) _ _ (pr1 f)).
- exact (nat_trans_ax (adjunit HG) _ _ (pr2 f)).
Qed.
Definition pair_equivalence_of_cats_unit
: functor_identity (category_binproduct C₁ C₂) ⟹ L ∙ R.
Proof.
use make_nat_trans.
- exact pair_equivalence_of_cats_unit_data.
- exact pair_equivalence_of_cats_unit_is_nat_trans.
Defined.
Definition pair_equivalence_of_cats_counit_data
: nat_trans_data (R ∙ L) (functor_identity _)
:= λ x, adjcounit HF (pr1 x) ,, adjcounit HG (pr2 x).
Definition pair_equivalence_of_cats_counit_is_nat_trans
: is_nat_trans
_ _
pair_equivalence_of_cats_counit_data.
Proof.
intros x y f.
use pathsdirprod ; cbn.
- exact (nat_trans_ax (adjcounit HF) _ _ (pr1 f)).
- exact (nat_trans_ax (adjcounit HG) _ _ (pr2 f)).
Qed.
Definition pair_equivalence_of_cats_counit
: R ∙ L ⟹ functor_identity _.
Proof.
use make_nat_trans.
- exact pair_equivalence_of_cats_counit_data.
- exact pair_equivalence_of_cats_counit_is_nat_trans.
Defined.
Definition pair_equivalence_of_cats
: equivalence_of_cats
(category_binproduct C₁ C₂)
(category_binproduct C₁' C₂').
Proof.
use make_equivalence_of_cats.
- use make_adjunction_data.
+ exact L.
+ exact R.
+ exact pair_equivalence_of_cats_unit.
+ exact pair_equivalence_of_cats_counit.
- split.
+ intro x.
use is_iso_binprod_iso.
× exact (iso_is_iso
(nat_iso_pointwise_iso
(unit_nat_iso_from_adj_equivalence_of_cats HF)
(pr1 x))).
× exact (iso_is_iso
(nat_iso_pointwise_iso
(unit_nat_iso_from_adj_equivalence_of_cats HG)
(pr2 x))).
+ intro x.
use is_iso_binprod_iso.
× exact (iso_is_iso
(nat_iso_pointwise_iso
(counit_nat_iso_from_adj_equivalence_of_cats HF)
(pr1 x))).
× exact (iso_is_iso
(nat_iso_pointwise_iso
(counit_nat_iso_from_adj_equivalence_of_cats HG)
(pr2 x))).
Defined.
End PairEquivalence.
Definition pair_adj_equivalence_of_cats
{C₁ C₁' C₂ C₂' : category}
{F : C₁ ⟶ C₁'}
{G : C₂ ⟶ C₂'}
(HF : adj_equivalence_of_cats F)
(HG : adj_equivalence_of_cats G)
: adj_equivalence_of_cats (pair_functor F G)
:= adjointificiation (pair_equivalence_of_cats HF HG).