Library UniMath.CategoryTheory.limits.cats.limits

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Contents :
Definition of limits

Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.

Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.MoreFoundations.Propositions.

Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.FunctorCategory.

Local Open Scope cat.

Section lim_def.

Definition cone {J C : precategory} (F : functor J C) (c : C) : UU :=
(f : (v : J), Cc,F v),
(u v : J) (e : Ju,v), f u · # F e = f v.

Definition make_cone {J C : precategory} {F : functor J C} {c : C}
(f : v, Cc, F v) (Hf : u v (e : Ju,v) , f u · # F e = f v) :
cone F c := tpair _ f Hf.

Definition coneOut {J C : precategory} {F : functor J C} {c : C} (cc : cone F c) :
v, Cc, F v := pr1 cc.

Lemma coneOutCommutes {J C : precategory} {F : functor J C} {c : C}
(cc : cone F c) : u v (e : Ju,v), coneOut cc u · # F e = coneOut cc v.
Proof.
apply (pr2 cc).
Qed.

Definition isLimCone {J C : precategory} (F : functor J C)
(l : C) (cc0 : cone F l) : UU := (c : C) (cc : cone F c),
iscontr ( x : Cc,l, v, x · coneOut cc0 v = coneOut cc v).

Definition LimCone {J C : precategory} (F : functor J C) : UU :=
(A : ( l, cone F l)), isLimCone F (pr1 A) (pr2 A).

Definition make_LimCone {J C : precategory} (F : functor J C)
(c : C) (cc : cone F c) (isCC : isLimCone F c cc) : LimCone F :=
tpair _ (tpair _ c cc) isCC.

Definition lim {J C : precategory} {F : functor J C} (CC : LimCone F) : C
:= pr1 (pr1 CC).

Definition limCone {J C : precategory} {F : functor J C} (CC : LimCone F) :
cone F (lim CC) := pr2 (pr1 CC).

Definition limOut {J C : precategory} {F : functor J C} (CC : LimCone F) :
v, Clim CC,F v := coneOut (limCone CC).

Lemma limOutCommutes {J C : precategory} {F : functor J C}
(CC : LimCone F) : u v (e : Ju,v),
limOut CC u · # F e = limOut CC v.
Proof.
exact (coneOutCommutes (limCone CC)).
Qed.

Lemma limUnivProp {J C : precategory} {F : functor J C}
(CC : LimCone F) : (c : C) (cc : cone F c),
iscontr ( x : Cc, lim CC, v, x · limOut CC v = coneOut cc v).
Proof.
exact (pr2 CC).
Qed.

Lemma isaprop_isLimCone {J C : precategory} (F : functor J C) (c0 : C)
(cc0 : cone F c0) : isaprop (isLimCone F c0 cc0).
Proof.
repeat (apply impred; intro).
apply isapropiscontr.
Qed.

Definition isLimCone_LimCone {J C : precategory} {F : functor J C}
(CC : LimCone F)
: isLimCone F (lim CC) (tpair _ (limOut CC) (limOutCommutes CC))
:= pr2 CC.

Definition limArrow {J C : precategory} {F : functor J C} (CC : LimCone F)
(c : C) (cc : cone F c) : Cc, lim CC := pr1 (pr1 (isLimCone_LimCone CC c cc)).

Lemma limArrowCommutes {J C : precategory} {F : functor J C} (CC : LimCone F)
(c : C) (cc : cone F c) u :
limArrow CC c cc · limOut CC u = coneOut cc u.
Proof.
exact ((pr2 (pr1 (isLimCone_LimCone CC _ cc))) u).
Qed.

Lemma limArrowUnique {J C : precategory} {F : functor J C} (CC : LimCone F)
(c : C) (cc : cone F c) (k : Cc, lim CC)
(Hk : u, k · limOut CC u = coneOut cc u) :
k = limArrow CC c cc.
Proof.
now apply path_to_ctr, Hk.
Qed.

Lemma Cone_precompose {J C : precategory} {F : functor J C}
{c : C} (cc : cone F c) (x : C) (f : Cx,c) :
u v (e : Ju,v), (f · coneOut cc u) · # F e = f · coneOut cc v.
Proof.
now intros u v e; rewrite <- assoc, coneOutCommutes.
Qed.

Lemma limArrowEta {J C : precategory} {F : functor J C} (CC : LimCone F)
(c : C) (f : Cc, lim CC) :
f = limArrow CC c (tpair _ (λ u, f · limOut CC u)
(Cone_precompose (limCone CC) c f)).
Proof.
now apply limArrowUnique.
Qed.

Definition limOfArrows {J C : precategory} {F1 F2 : functor J C}
(CC1 : LimCone F1) (CC2 : LimCone F2)
(f : u, CF1 u,F2 u)
(fNat : u v (e : Ju,v), f u · # F2 e = # F1 e · f v) :
Clim CC1 , lim CC2.
Proof.
apply limArrow; use make_cone.
- now intro u; apply (limOut CC1 u · f u).
- abstract (intros u v e; simpl;
now rewrite <- assoc, fNat, assoc, limOutCommutes).
Defined.

Lemma limOfArrowsOut {J C : precategory} {F1 F2 : functor J C}
(CC1 : LimCone F1) (CC2 : LimCone F2)
(f : u, CF1 u,F2 u)
(fNat : u v (e : Ju,v), f u · # F2 e = # F1 e · f v) :
u, limOfArrows CC1 CC2 f fNat · limOut CC2 u =
limOut CC1 u · f u.
Proof.
now unfold limOfArrows; intro u; rewrite limArrowCommutes.
Qed.

Lemma postCompWithLimOfArrows_subproof
{J C : precategory} {F1 F2 : functor J C}
(CC1 : LimCone F1) (CC2 : LimCone F2)
(f : u, CF1 u,F2 u)
(fNat : u v (e : Ju,v), f u · # F2 e = # F1 e · f v)
(x : C) (cc : cone F1 x) u v (e : Ju,v) :
(coneOut cc u · f u) · # F2 e = coneOut cc v · f v.
Proof.
now rewrite <- (coneOutCommutes cc u v e), <- assoc, fNat, assoc.
Defined.

Lemma postcompWithLimOfArrows
{J C : precategory} {F1 F2 : functor J C}
(CC1 : LimCone F1) (CC2 : LimCone F2)
(f : u, CF1 u,F2 u)
(fNat : u v (e : Ju,v), f u · # F2 e = # F1 e · f v)
(x : C) (cc : cone F1 x) :
limArrow CC1 x cc · limOfArrows CC1 CC2 f fNat =
limArrow CC2 x (make_cone (λ u, coneOut cc u · f u)
(postCompWithLimOfArrows_subproof CC1 CC2 f fNat x cc)).
Proof.
apply limArrowUnique; intro u.
now rewrite <- assoc, limOfArrowsOut, assoc, limArrowCommutes.
Qed.

Lemma postcompWithLimArrow {J C : precategory} {F : functor J C}
(CC : LimCone F) (c : C) (cc : cone F c) (d : C) (k : Cd,c) :
k · limArrow CC c cc =
limArrow CC d (make_cone (λ u, k · coneOut cc u)
(Cone_precompose cc d k)).
Proof.
apply limArrowUnique.
now intro u; rewrite <- assoc, limArrowCommutes.
Qed.

Lemma lim_endo_is_identity {J C : precategory} {F : functor J C}
(CC : LimCone F) (k : lim CC --> lim CC)
(H : u, k · limOut CC u = limOut CC u) :
identity _ = k.
Proof.
use (uniqueExists (limUnivProp CC _ _)).
- now apply (limCone CC).
- now intros v; apply id_left.
- simpl; now apply H.
Qed.

Definition iso_from_lim_to_lim {J C : precategory} {F : functor J C}
(CC CC' : LimCone F) : iso (lim CC) (lim CC').
Proof.
use make_iso.
- apply limArrow, limCone.
- use is_iso_qinv.
+ apply limArrow, limCone.
+ abstract (now split; apply pathsinv0, lim_endo_is_identity; intro u;
rewrite <- assoc, limArrowCommutes; eapply pathscomp0; try apply limArrowCommutes).
Defined.

End lim_def.

Section Lims.

Definition Lims (C : precategory) : UU := (J : precategory) (F : functor J C), LimCone F.
Definition hasLims : UU :=
(J C : precategory) (F : functor J C), ishinh (LimCone F).
Definition Lims_of_shape (J C : precategory) : UU := (F : functor J C), LimCone F.

Section Universal_Unique.

Context (C : univalent_category).

Let H : is_univalent C := pr2 C.

Lemma isaprop_Lims: isaprop (Lims C).
Proof.
apply impred; intro J; apply impred; intro F.
apply invproofirrelevance; intros Hccx Hccy.
apply subtypePath.
- intro; apply isaprop_isLimCone.
- apply (total2_paths_f (isotoid _ H (iso_from_lim_to_lim Hccx Hccy))).
set (B c := v, Cc,F v).
set (C' (c : C) f := u v (e : Ju,v), @compose _ c _ _ (f u) (# F e) = f v).
rewrite (@transportf_total2 _ B C').
apply subtypePath.
+ intro; repeat (apply impred; intro); apply (pr2 H).
+ abstract (now simpl; eapply pathscomp0; [apply transportf_isotoid_dep'|];
apply funextsec; intro v; rewrite inv_isotoid, idtoiso_isotoid;
cbn; unfold precomp_with; rewrite id_right; apply limArrowCommutes).
Qed.

End Universal_Unique.
End Lims.

Section LimFunctor.

Variable A C : precategory.
Variable hsC : has_homsets C.
Variable (J : precategory).
Variable (D : functor J [A, C, hsC]).

Definition functor_pointwise (a : A) : functor J C.
Proof.
use tpair.
- apply (tpair _ (λ v, pr1 (D v) a)).
intros u v e; simpl; apply (pr1 (# D e) a).
- abstract (use tpair;
[ intro x; simpl;
apply (toforallpaths _ _ _ (maponpaths pr1 (functor_id D x)) a)
| intros x y z f g; simpl;
apply (toforallpaths _ _ _ (maponpaths pr1 (functor_comp D f g)) a)]).
Defined.

Variable (HCg : (a : A), LimCone (functor_pointwise a)).

Definition LimFunctor_ob (a : A) : C := lim (HCg a).

Definition LimFunctor_mor (a a' : A) (f : Aa, a') :
CLimFunctor_ob a,LimFunctor_ob a'.
Proof.
use limOfArrows.
- now intro u; apply (# (pr1 (D u)) f).
- abstract (now intros u v e; simpl; apply (nat_trans_ax (# D e))).
Defined.

Definition LimFunctor_data : functor_data A C := tpair _ _ LimFunctor_mor.

Lemma is_functor_LimFunctor_data : is_functor LimFunctor_data.
Proof.
split.
- intro a; simpl.
apply pathsinv0, lim_endo_is_identity; intro u.
unfold LimFunctor_mor; rewrite limOfArrowsOut.
assert (H : # (pr1 (D u)) (identity a) = identity (pr1 (D u) a)).
apply (functor_id (D u) a).
now rewrite H, id_right.
- intros a b c fab fbc; simpl; unfold LimFunctor_mor.
apply pathsinv0.
eapply pathscomp0; [now apply postcompWithLimOfArrows|].
apply pathsinv0, limArrowUnique; intro u.
rewrite limOfArrowsOut, (functor_comp (D u)); simpl.
now rewrite <- assoc.
Qed.

Definition LimFunctor : functor A C := tpair _ _ is_functor_LimFunctor_data.

Definition lim_nat_trans_in_data v : [A, C, hsC] LimFunctor, D v .
Proof.
use tpair.
- intro a; exact (limOut (HCg a) v).
- abstract (intros a a' f; apply (limOfArrowsOut (HCg a) (HCg a'))).
Defined.

Definition cone_pointwise (F : [A,C,hsC]) (cc : cone D F) a :
cone (functor_pointwise a) (pr1 F a).
Proof.
use make_cone.
- now intro v; apply (pr1 (coneOut cc v) a).
- abstract (intros u v e;
now apply (nat_trans_eq_pointwise (coneOutCommutes cc u v e))).
Defined.

Lemma LimFunctor_unique (F : [A, C, hsC]) (cc : cone D F) :
iscontr ( x : [A, C, hsC] F, LimFunctor ,
v, x · lim_nat_trans_in_data v = coneOut cc v).
Proof.
use tpair.
- use tpair.
+ apply (tpair _ (λ a, limArrow (HCg a) _ (cone_pointwise F cc a))).
abstract (intros a a' f; simpl; apply pathsinv0; eapply pathscomp0;
[ apply (postcompWithLimOfArrows (HCg a))
| apply pathsinv0; eapply pathscomp0;
[ apply postcompWithLimArrow
| apply limArrowUnique; intro u; eapply pathscomp0;
[ now apply limArrowCommutes | now use nat_trans_ax]]]).
+ abstract (intro u; apply (nat_trans_eq hsC); simpl; intro a;
now apply (limArrowCommutes (HCg a))).
- abstract (intro t; destruct t as [t1 t2];
apply subtypePath; simpl;
[ intro; apply impred; intro u; apply functor_category_has_homsets
| apply (nat_trans_eq hsC); simpl; intro a;
apply limArrowUnique; intro u;
now apply (nat_trans_eq_pointwise (t2 u))]).
Defined.

Lemma LimFunctorCone : LimCone D.
Proof.
use make_LimCone.
- exact LimFunctor.
- use make_cone.
+ now apply lim_nat_trans_in_data.
+ abstract (now intros u v e; apply (nat_trans_eq hsC);
intro a; apply (limOutCommutes (HCg a))).
- now intros F cc; simpl; apply (LimFunctor_unique _ cc).
Defined.

End LimFunctor.

Lemma LimsFunctorCategory (A C : precategory) (hsC : has_homsets C)
(HC : Lims C) : Lims [A,C,hsC].
Proof.
now intros g d; apply LimFunctorCone.
Defined.

Lemma LimsFunctorCategory_of_shape (J A C : precategory) (hsC : has_homsets C)
(HC : Lims_of_shape J C) : Lims_of_shape J [A,C,hsC].
Proof.
now intros d; apply LimFunctorCone.
Defined.