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Require Import MyTacs HoTT.
Require Import colimit.
Set Implicit Arguments.
Context `{fs : Funext}.
Context `{ua : Univalence}.
Section cocone_product_r.
Variable G:graph.
Variable D: diagram G.
Variable A: Type.
Definition pdt_diagram_r : diagram G.
refine (Build_diagram _ _ _).
+ intros i. exact ((D i) * A).
+ simpl. intros i j f x. exact (diagram1 D f (fst x), snd x).
Defined.
Definition pdt_cocone_r {Q: Type} (C: cocone D Q) : cocone pdt_diagram_r (Q * A).
unfold cocone in *.
refine (exist _ _ _). simpl in *.
intros i z. exact (C.1 i (fst z) , snd z).
intros i j f z. simpl.
exact (path_prod' (C.2 i j f (fst z)) 1).
Defined.
Lemma cocone_product_r : forall (X:Type), cocone pdt_diagram_r X <~> cocone D (A -> X).
intros X.
refine (equiv_adjointify _ _ _ _).
+ intros C. unfold cocone in *.
refine (exist _ _ _).
intros i x a. exact (C.1 i (x, a)).
intros i j f x. simpl. apply path_forall; intros a.
exact (C.2 _ _ _ (x,a)).
+ intros C. unfold cocone in *.
refine (exist _ _ _).
intros i z. exact (C.1 i (fst z) (snd z)).
intros i j f z. simpl.
f_ap. exact (C.2 _ _ _ _).
+ intros [q pp]. simpl.
refine (path_sigma _ _ _ _ _).
- reflexivity.
- simpl. repeat (apply path_forall; intro). apply eta_path_forall.
+ intros [q pp]. simpl.
refine (path_sigma _ _ _ _ _).
- reflexivity.
- simpl. repeat (apply path_forall; intro). unfold path_forall.
rewrite eisretr. reflexivity.
Defined.
Lemma colimit_product_r (Q:Type) (C: cocone D Q) (H: is_colimit C) : is_colimit (pdt_cocone_r C).
unfold is_colimit. intros X.
assert (H': map_to_cocone (pdt_cocone_r C) X = ((cocone_product_r X)^-1) o (map_to_cocone C (A -> X)) o ((equiv_uncurry Q _ X)^-1)).
+ apply path_forall; intros F.
refine (path_sigma _ _ _ _ _).
- reflexivity.
- simpl.
apply path_forall; intros i.
apply path_forall; intros j.
apply path_forall; intros f.
apply path_forall; intros z.
destruct (C.2 i j f (fst z)). simpl.
reflexivity.
+ rewrite H'. refine isequiv_compose.
refine isequiv_compose. apply H.
Defined.
End cocone_product_r.
Section cocone_product_l.
Variable G:graph.
Variable D: diagram G.
Variable A: Type.
Definition pdt_diagram_l : diagram G.
refine (Build_diagram _ _ _).
+ intros i. exact (A * (D i)).
+ simpl. intros i j f x. exact (fst x, diagram1 D f (snd x)).
Defined.
Definition pdt_cocone_l {Q: Type} (C: cocone D Q) : cocone pdt_diagram_l (A*Q).
unfold cocone in *.
refine (exist _ _ _). simpl in *.
intros i z. exact (fst z, C.1 i (snd z)).
intros i j f z. simpl.
f_ap. exact (C.2 _ _ _ _).
Defined.
Lemma cocone_product_l : forall (X:Type), cocone pdt_diagram_l X <~> cocone D (A -> X).
intros X.
refine (equiv_adjointify _ _ _ _).
+ intros C. unfold cocone in *.
refine (exist _ _ _).
intros i x a. exact (C.1 i (a, x)).
intros i j f x. simpl. apply path_forall; intros a.
exact (C.2 _ _ _ (a,x)).
+ intros C. unfold cocone in *.
refine (exist _ _ _).
intros i z. exact (C.1 i (snd z) (fst z)).
intros i j f z. simpl.
f_ap. exact (C.2 _ _ _ _).
+ intros [q pp]. simpl.
refine (path_sigma _ _ _ _ _).
- reflexivity.
- simpl. repeat (apply path_forall; intro). apply eta_path_forall.
+ intros [q pp]. simpl.
refine (path_sigma _ _ _ _ _).
- reflexivity.
- simpl. repeat (apply path_forall; intro). unfold path_forall.
rewrite eisretr. reflexivity.
Defined.
Lemma colimit_product_l (Q:Type) (C: cocone D Q) (H: is_colimit C) : is_colimit (pdt_cocone_l C).
unfold is_colimit. intros X.
assert (H': map_to_cocone (pdt_cocone_l C) X = ((cocone_product_l X)^-1) o (map_to_cocone C (A -> X)) o ((equiv_uncurry Q A X)^-1) o (λ f a, f (snd a,fst a))).
+ apply path_forall; intros F.
refine (path_sigma _ _ _ _ _).
- reflexivity.
- simpl.
apply path_forall; intros i.
apply path_forall; intros j.
apply path_forall; intros f.
apply path_forall; intros z.
destruct (C.2 i j f (snd z)). simpl.
reflexivity.
+ rewrite H'. refine isequiv_compose.
refine isequiv_compose. refine isequiv_compose.
refine (isequiv_adjointify (λ (x : A ∧ Q → X) (a : Q ∧ A), x (snd a, fst a)) (λ (x : Q ∧ A → X) (a : A ∧ Q), x (snd a, fst a)) _ _).
{ intro x. reflexivity. }
{ intro x. reflexivity. }
apply H.
Defined.
End cocone_product_l.
Section colimit_unicity.
Definition postcompose {A X Y: Type} (f: X->Y) : (A->X) -> (A->Y) :=
fun g => f o g.
Lemma yoneda_map {A B: Type} (F: forall X, (A->X) <~> (B->X))
(H: forall (X Y: Type) (i: X->Y), (postcompose i) o (F X) == (F Y) o (postcompose i))
: IsEquiv ((F B)^-1 idmap). (* A <~> B *)
set (f := (F B)^-1 idmap). set (g := (F A) idmap).
refine (isequiv_adjointify _ g _ _).
+ specialize (H _ _ f idmap). unfold postcompose in H.
apply ap10. rewrite H.
apply eisretr.
+ specialize (H _ _ g f). apply ap10.
apply (equiv_inj (F A)). unfold postcompose in H.
rewrite <- H. unfold f. rewrite (eisretr (F B)). reflexivity.
Defined.
Lemma map_to_cocone_postcompose {G: graph} {D: diagram G} {P: Type} (C: cocone D P) {X Y: Type} (f: X->Y)
: (map_to_cocone C Y) o (postcompose f) == (λ C, map_to_cocone C _ f) o (map_to_cocone C X).
intros g. destruct C as [q pq].
refine (path_cocone _ _).
+ intros i x. reflexivity.
+ unfold postcompose. simpl.
intros i j f0 x. hott_simpl. apply ap_compose.
Defined.
Lemma colimit_unicity (G: graph) (D: diagram G) (P Q: Type) (C: cocone D P) (C': cocone D Q) (colimP : is_colimit C) (colimQ : is_colimit C')
: IsEquiv (@equiv_inv _ _ _ (colimP Q) C'). (* P <~> Q *)
unfold is_colimit in *.
set (φP := map_to_cocone C) in *.
set (φQ := map_to_cocone C') in *.
assert (C' = (φQ Q idmap)).
{ refine (path_cocone _ _).
intros i x. reflexivity.
intros i j f x. simpl. hott_simpl. }
rewrite X. clear X.
set (F := λ X, BuildEquiv _ _ ((φQ X)^-1 o (φP X)) _).
assert ((φP Q)^-1 (φQ Q idmap) = (F Q)^-1 idmap). reflexivity.
rewrite X. clear X.
refine (yoneda_map F _).
intros X Y i q. simpl.
transitivity ((φQ Y)^-1 (map_to_cocone (φP X q) _ i)).
- assert (H: forall C, postcompose i ((φQ X)^-1 C) = (φQ Y)^-1 (map_to_cocone C _ i)); [|apply H].
clear F colimP φP C. intros C.
apply (equiv_inj (φQ Y)). rewrite eisretr.
specialize (map_to_cocone_postcompose C' i ((φQ X)^-1 C)); intros H.
rewrite eisretr in H. assumption.
- apply ap. symmetry. apply map_to_cocone_postcompose.
Defined.
End colimit_unicity.