Library TypeTheory.Auxiliary.Auxiliary
TypeTheory.Auxiliary.Auxiliary
Part of the TypeTheory library (Ahrens, Lumsdaine, Voevodsky, 2015–present).
Auxiliary background lemmas for the Ahrens/Lumsdaine/Voevodsky “Systems” project.
Possibly some should be upstreamed to “UniMath” eventually.
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.Combinatorics.StandardFiniteSets.
Require Import UniMath.CategoryTheory.limits.graphs.limits.
Require Import UniMath.CategoryTheory.limits.graphs.pullbacks.
Require Import UniMath.CategoryTheory.categories.category_hset_structures.
Require Import TypeTheory.Auxiliary.CategoryTheoryImports.
Require Import TypeTheory.Auxiliary.UnicodeNotations.
Set Automatic Introduction.
Undelimit Scope transport.
Notation "( x , y , .. , z )" := (dirprodpair .. (dirprodpair x y) .. z) : core_scope.
Replaces builtin notation for pair, since we use dirprod, dirprodpair instead of prod, pair.
Redeclare this notation, along with a new scope.
Notation "ff ;; gg" := (compose ff gg)
(at level 50, left associativity, format "ff ;; gg")
: mor_scope.
Delimit Scope mor_scope with mor.
Bind Scope mor_scope with precategory_morphisms.
Open Scope mor_scope.
(at level 50, left associativity, format "ff ;; gg")
: mor_scope.
Delimit Scope mor_scope with mor.
Bind Scope mor_scope with precategory_morphisms.
Open Scope mor_scope.
Tactic Notation "etrans" := eapply pathscomp0.
Tactic Notation "rew_trans_@" := repeat (etrans ; [ apply transport_f_f |]).
Tactic Notation "sym" := apply pathsinv0.
Tactic Notation "assoc" := apply @pathsinv0, path_assoc.
Tactic Notation "cancel_postcomposition" := apply cancel_postcomposition.
Arguments functor_on_inv_from_iso {_ _} _ {_ _} f.
Path-algebra: general lemmas about transport, equivalences, etc.
Definition maponpaths_2 {X Y Z : UU} (f : X -> Y -> Z) {x x'} (e : x = x') y
: f x y = f x' y
:= maponpaths (fun x => f x y) e.
: f x y = f x' y
:= maponpaths (fun x => f x y) e.
TODO: see https://github.com/UniMath/UniMath/issues/677
Lemma pr1_issurjective' {X : UU} {P : X -> UU} :
(∏ x : X, ∥ P x ∥) -> issurjective (pr1 : (∑ x, P x) -> X).
Proof.
intros ne x. simple refine (hinhuniv _ (ne x)).
intros p. apply hinhpr.
exact ((x,,p),,idpath _).
Defined.
Lemma fibers_inhab_if_pr1_issurjective {X : UU} {P : X -> UU} :
(∏ x : X, ∥ P x ∥) <- issurjective (pr1 : (∑ x, P x) -> X).
Proof.
intros ne x. simple refine (hinhuniv _ (ne x)).
intros p. apply hinhpr.
cbn in p.
destruct p as [[a b] c].
cbn in *.
induction c.
assumption.
Defined.
Lemma isaprop_fiber_if_isinclpr1
: ∏ (X : UU) (isasetX : isaset X) (P : X → UU), (∏ x : X, isaprop (P x)) <- isincl (pr1 : (∑ x, P x) -> X).
Proof.
intros X isasetX P H x.
unfold isincl in H. unfold isofhlevelf in H.
apply invproofirrelevance.
intros p p'.
assert (X0 : x,,p = x,,p').
{ specialize (H x).
assert (H1 : (x,,p),, idpath _ = ((x,,p'),,idpath _ : hfiber pr1 x)).
{ apply proofirrelevance. apply H. }
apply (base_paths _ _ H1).
}
set (XR := fiber_paths X0). cbn in XR.
etrans. Focus 2. apply XR.
apply pathsinv0.
etrans. apply maponpaths_2. apply (isasetX _ _ _ (idpath x)).
apply idpath_transportf.
Defined.
Lemma weqhomot {A B : UU} (f : A -> B) (w : A ≃ B) (H : w ~ f) : isweq f.
Proof.
apply isweqhomot with w. apply H. apply weqproperty.
Defined.
Lemma pr1_transportf (A : UU) (B : A -> UU) (P : ∏ a, B a -> UU)
(a a' : A) (e : a = a') (xs : ∑ b : B a, P _ b):
pr1 (transportf (fun x => ∑ b : B x, P _ b) e xs) =
transportf (fun x => B x) e (pr1 xs).
Proof.
destruct e; apply idpath.
Defined.
Lemma transportf_const (A B : UU) (a a' : A) (e : a = a') (b : B) :
transportf (fun _ => B) e b = b.
Proof.
induction e.
apply idpath.
Defined.
Lemma transportf_forall {A B} (C : A -> B -> UU)
{x0 x1 : A} (e : x0 = x1) (f : forall y:B, C x0 y)
: transportf (fun x => forall y, C x y) e f
= fun y => transportf (fun x => C x y) e (f y).
Proof.
destruct e; apply idpath.
Defined.
Definition transportf_forall_var :
∏ (A : UU) (B : A -> UU) (C : UU)
(a1 a2 : A) (e : a1 = a2)
(f : B a1 -> C),
transportf (λ x : A, ∏ y : B x, C) e f =
(λ y : B a2 , f (transportb B e y)).
Proof.
intros A B D a1 a2 e f.
induction e.
apply idpath.
Defined.
Definition transportf_forall_var2 :
∏ (A : UU) (B C : A -> UU)
(a1 a2 : A) (e : a1 = a2)
(f : B a1 -> C a1),
transportf (λ x : A, ∏ y : B x, C x) e f =
(λ y : B a2 , transportf _ e (f (transportb B e y))).
Proof.
intros A B D a1 a2 e f.
induction e.
apply idpath.
Defined.
Lemma maponpaths_apply {A B} {f0 f1 : A -> B} (e : f0 = f1) (x : A)
: maponpaths (fun f => f x) e
= toforallpaths _ _ _ e x.
Proof.
destruct e; apply idpath.
Defined.
Lemma maponpaths_eq_idpath
: ∏ (T1 T2 : UU) (f : T1 -> T2) (t1 : T1) (e : t1 = t1)
(H : e = idpath _ ), maponpaths f e = idpath _ .
Proof.
intros.
exact (maponpaths (maponpaths f) H).
Defined.
Lemma transportf_comp_lemma (X : UU) (B : X -> UU) {A A' A'': X} (e : A = A'') (e' : A' = A'')
(x : B A) (x' : B A')
: transportf _ (e @ !e') x = x'
-> transportf _ e x = transportf _ e' x'.
Proof.
intro H.
eapply pathscomp0. Focus 2.
apply maponpaths. exact H.
eapply pathscomp0. Focus 2.
symmetry. apply transport_f_f.
apply (maponpaths (fun p => transportf _ p x)).
apply pathsinv0.
eapply pathscomp0.
- apply @pathsinv0, path_assoc.
- eapply pathscomp0.
apply maponpaths.
apply pathsinv0l.
apply pathscomp0rid.
Defined.
Lemma transportf_comp_lemma_hset (X : UU) (B : X -> UU) (A : X) (e : A = A)
{x x' : B A} (hs : isaset X)
: x = x'
-> transportf _ e x = x'.
Proof.
intros ex.
apply @pathscomp0 with (transportf _ (idpath _) x).
- apply (maponpaths (fun p => transportf _ p x)).
apply hs.
- exact ex.
Qed.
Lemma transportf_pair {A B} (P : A × B -> UU) {a a' : A} {b b' : B}
(eA : a = a') (eB : b = b') (p : P (a,,b))
: transportf P (pathsdirprod eA eB) p =
transportf (fun bb => P(a',,bb) ) eB (transportf (fun aa => P(aa,,b)) eA p).
Proof.
induction eA. induction eB. apply idpath.
Defined.
Lemma transportf_ext (X : UU) (B : X -> UU) (A A' : X) (e e' : A = A') p :
e = e' -> transportf _ e p = transportf B e' p.
Proof.
intro H; induction H; apply idpath.
Defined.
(∏ x : X, ∥ P x ∥) -> issurjective (pr1 : (∑ x, P x) -> X).
Proof.
intros ne x. simple refine (hinhuniv _ (ne x)).
intros p. apply hinhpr.
exact ((x,,p),,idpath _).
Defined.
Lemma fibers_inhab_if_pr1_issurjective {X : UU} {P : X -> UU} :
(∏ x : X, ∥ P x ∥) <- issurjective (pr1 : (∑ x, P x) -> X).
Proof.
intros ne x. simple refine (hinhuniv _ (ne x)).
intros p. apply hinhpr.
cbn in p.
destruct p as [[a b] c].
cbn in *.
induction c.
assumption.
Defined.
Lemma isaprop_fiber_if_isinclpr1
: ∏ (X : UU) (isasetX : isaset X) (P : X → UU), (∏ x : X, isaprop (P x)) <- isincl (pr1 : (∑ x, P x) -> X).
Proof.
intros X isasetX P H x.
unfold isincl in H. unfold isofhlevelf in H.
apply invproofirrelevance.
intros p p'.
assert (X0 : x,,p = x,,p').
{ specialize (H x).
assert (H1 : (x,,p),, idpath _ = ((x,,p'),,idpath _ : hfiber pr1 x)).
{ apply proofirrelevance. apply H. }
apply (base_paths _ _ H1).
}
set (XR := fiber_paths X0). cbn in XR.
etrans. Focus 2. apply XR.
apply pathsinv0.
etrans. apply maponpaths_2. apply (isasetX _ _ _ (idpath x)).
apply idpath_transportf.
Defined.
Lemma weqhomot {A B : UU} (f : A -> B) (w : A ≃ B) (H : w ~ f) : isweq f.
Proof.
apply isweqhomot with w. apply H. apply weqproperty.
Defined.
Lemma pr1_transportf (A : UU) (B : A -> UU) (P : ∏ a, B a -> UU)
(a a' : A) (e : a = a') (xs : ∑ b : B a, P _ b):
pr1 (transportf (fun x => ∑ b : B x, P _ b) e xs) =
transportf (fun x => B x) e (pr1 xs).
Proof.
destruct e; apply idpath.
Defined.
Lemma transportf_const (A B : UU) (a a' : A) (e : a = a') (b : B) :
transportf (fun _ => B) e b = b.
Proof.
induction e.
apply idpath.
Defined.
Lemma transportf_forall {A B} (C : A -> B -> UU)
{x0 x1 : A} (e : x0 = x1) (f : forall y:B, C x0 y)
: transportf (fun x => forall y, C x y) e f
= fun y => transportf (fun x => C x y) e (f y).
Proof.
destruct e; apply idpath.
Defined.
Definition transportf_forall_var :
∏ (A : UU) (B : A -> UU) (C : UU)
(a1 a2 : A) (e : a1 = a2)
(f : B a1 -> C),
transportf (λ x : A, ∏ y : B x, C) e f =
(λ y : B a2 , f (transportb B e y)).
Proof.
intros A B D a1 a2 e f.
induction e.
apply idpath.
Defined.
Definition transportf_forall_var2 :
∏ (A : UU) (B C : A -> UU)
(a1 a2 : A) (e : a1 = a2)
(f : B a1 -> C a1),
transportf (λ x : A, ∏ y : B x, C x) e f =
(λ y : B a2 , transportf _ e (f (transportb B e y))).
Proof.
intros A B D a1 a2 e f.
induction e.
apply idpath.
Defined.
Lemma maponpaths_apply {A B} {f0 f1 : A -> B} (e : f0 = f1) (x : A)
: maponpaths (fun f => f x) e
= toforallpaths _ _ _ e x.
Proof.
destruct e; apply idpath.
Defined.
Lemma maponpaths_eq_idpath
: ∏ (T1 T2 : UU) (f : T1 -> T2) (t1 : T1) (e : t1 = t1)
(H : e = idpath _ ), maponpaths f e = idpath _ .
Proof.
intros.
exact (maponpaths (maponpaths f) H).
Defined.
Lemma transportf_comp_lemma (X : UU) (B : X -> UU) {A A' A'': X} (e : A = A'') (e' : A' = A'')
(x : B A) (x' : B A')
: transportf _ (e @ !e') x = x'
-> transportf _ e x = transportf _ e' x'.
Proof.
intro H.
eapply pathscomp0. Focus 2.
apply maponpaths. exact H.
eapply pathscomp0. Focus 2.
symmetry. apply transport_f_f.
apply (maponpaths (fun p => transportf _ p x)).
apply pathsinv0.
eapply pathscomp0.
- apply @pathsinv0, path_assoc.
- eapply pathscomp0.
apply maponpaths.
apply pathsinv0l.
apply pathscomp0rid.
Defined.
Lemma transportf_comp_lemma_hset (X : UU) (B : X -> UU) (A : X) (e : A = A)
{x x' : B A} (hs : isaset X)
: x = x'
-> transportf _ e x = x'.
Proof.
intros ex.
apply @pathscomp0 with (transportf _ (idpath _) x).
- apply (maponpaths (fun p => transportf _ p x)).
apply hs.
- exact ex.
Qed.
Lemma transportf_pair {A B} (P : A × B -> UU) {a a' : A} {b b' : B}
(eA : a = a') (eB : b = b') (p : P (a,,b))
: transportf P (pathsdirprod eA eB) p =
transportf (fun bb => P(a',,bb) ) eB (transportf (fun aa => P(aa,,b)) eA p).
Proof.
induction eA. induction eB. apply idpath.
Defined.
Lemma transportf_ext (X : UU) (B : X -> UU) (A A' : X) (e e' : A = A') p :
e = e' -> transportf _ e p = transportf B e' p.
Proof.
intro H; induction H; apply idpath.
Defined.
Lemma invmap_eq {A B : UU} (f : A ≃ B) (b : B) (a : A)
: b = f a → invmap f b = a.
Proof.
intro H.
apply (invmaponpathsweq f).
etrans. apply homotweqinvweq. apply H.
Defined.
Definition isweqpathscomp0l {X : UU} {x x' : X} (x'' : X) (e: x = x') :
isweq (fun (e' : x' = x'') => e @ e').
Proof.
intros.
apply (gradth _ (fun e'' => !e @ e'')).
- intro p. rewrite path_assoc. rewrite pathsinv0l.
apply idpath.
- intro p. rewrite path_assoc. rewrite pathsinv0r.
apply idpath.
Defined.
Definition weqpathscomp0l {X : UU} {x x'} (x'' : X) (e : x = x')
: (x' = x'') ≃ (x = x'').
Proof.
exact (_ ,, isweqpathscomp0l x'' e).
Defined.
Definition weqpathscomp0r {X : UU} (x:X) {x' x''} (e' : x' = x'')
: (x = x') ≃ (x = x'').
Proof.
exact (_ ,, isweqpathscomp0r x e').
Defined.
Definition weq_exchange_args {A B} (C : A -> B -> Type)
: (∏ a b, C a b) ≃ (∏ b a, C a b).
Proof.
use weqgradth.
- intros f b a; exact (f a b).
- intros g a b; exact (g b a).
- intros f; apply idpath.
- intros g; apply idpath.
Defined.
Definition isweqbandfmap_var {X Y : UU} (w : X -> Y)
(P : X → UU) (Q : Y → UU)
(fw : ∏ x : X, P x -> Q (w x))
: isweq w -> (∏ x, isweq (fw x)) -> isweq (bandfmap w P Q (λ x : X, fw x)).
Proof.
intros Hw Hfw.
apply (isweqbandfmap (weqpair w Hw) _ _ (fun x => weqpair _ (Hfw x))).
Defined.
Lemma weqtotal2asstol' {X : UU} (P : X → UU) (Q : forall x, P x → UU)
: (∑ (x : X) (p : P x), Q x p) ≃ (∑ (y : ∑ x, P x), Q (pr1 y) (pr2 y)).
Proof.
exact (weqtotal2asstol P (fun y => Q (pr1 y) (pr2 y))).
Defined.
Lemma weqtotal2asstor' {X : UU} (P : X → UU) (Q : forall x, P x → UU)
: (∑ (y : ∑ x, P x), Q (pr1 y) (pr2 y)) ≃ (∑ (x : X) (p : P x), Q x p).
Proof.
exact (weqtotal2asstor P (fun y => Q (pr1 y) (pr2 y))).
Defined.
Lemma issurjective_hinhpr (A : UU) : issurjective (@hinhpr A).
Proof.
intro a.
apply (squash_to_prop a).
- apply propproperty.
- intro aa. apply hinhpr.
exists aa.
apply proofirrelevance.
apply propproperty.
Defined.
Lemma issurjective_bandfmap {X Y : UU} (f : X → Y) (P : X → UU) (Q : Y → UU)
(fx : ∏ x : X, P x → Q (f x))
(Hf : issurjective f)
(Hfx : ∏ x, issurjective (fx x))
: issurjective (bandfmap f _ _ fx).
Proof.
intros [y q].
apply (squash_to_prop (Hf y)).
{ apply propproperty. }
intros [x Hx].
induction Hx.
apply (squash_to_prop (Hfx _ q)).
{ apply propproperty. }
intros [p Hp].
destruct Hp.
apply hinhpr.
exists (x,,p).
apply idpath.
Defined.
Lemma eq_equiv_from_retraction {A} {P : A -> A -> UU}
(w : forall a b, a = b -> P a b)
(v : forall a b, P a b -> a = b)
: (forall a b (p : P a b), w _ _ (v _ _ p) = p)
-> forall a b, isweq (w a b).
Proof.
intros wv a.
apply isweqtotaltofib. use gradth.
- intros bp. exists (pr1 bp). apply v, (pr2 bp).
- intros be; apply connectedcoconusfromt. - intros bp. use total2_paths_f. apply idpath. apply wv.
Qed.
Definition truncation_weq (A : UU) (is : isaprop A) : A ≃ ∥ A ∥.
Proof.
apply weqimplimpl.
- apply hinhpr.
- intro a. use (squash_to_prop a is). apply idfun.
- apply is.
- apply propproperty.
Defined.
Lemma is_iso_comp_is_iso {C : precategory} {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 (isopair f Hf) (isopair g Hg)).
Defined.
Lemma functor_is_iso_is_iso {C C' : precategory} (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 _ _ (isopair f fH)).
Defined.
Lemma inv_from_iso_from_is_z_iso {D: precategory} {a b : D}
(f: a --> b) (g : b --> a) (H : is_inverse_in_precat f g)
: inv_from_iso (f ,, (is_iso_from_is_z_iso _ (g ,, H)))
= g.
Proof.
cbn. apply id_right.
Qed.
The total type of morphisms of a precategory
Definition mor_total (C : precategory) : UU
:= ∑ (ab : C × C), C⟦pr2 ab, pr1 ab⟧.
Definition morphism_as_total {C : precategory} {a b : C} (f : a --> b) : mor_total C.
Proof.
exists (b,,a).
exact f.
Defined.
Definition source {C} (X : mor_total C) : C := pr2 (pr1 X).
Definition target {C} (X : mor_total C) : C := pr1 (pr1 X).
Definition morphism_from_total {C} (X : mor_total C)
: C ⟦source X, target X⟧
:= pr2 X.
Coercion morphism_from_total : mor_total >-> precategory_morphisms.
Definition functor_on_mor_total {C D : precategory} (F : functor C D)
(p : mor_total C) : mor_total D.
Proof.
exists (F (pr1 (pr1 p)) ,, F (pr2 (pr1 p)) ).
exact (#F p).
Defined.
Definition isweq_left_adj_equivalence_on_mor_total {C D : precategory} (F : functor C D)
(isC : is_univalent C) (isD : is_univalent D)
(H : adj_equivalence_of_precats F)
: isweq (functor_on_mor_total F).
Proof.
use (gradth _ _ _ _ ).
- apply (functor_on_mor_total (adj_equivalence_inv H)).
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (unit_pointwise_iso_from_adj_equivalence H).
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (unit_pointwise_iso_from_adj_equivalence H).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : C × C, C ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with. rewrite id_right.
rewrite assoc. assert (XR := nat_trans_ax (unit_from_are_adjoints (pr2 (pr1 H)))).
cbn in XR. rewrite <- XR.
rewrite <- assoc.
etrans. apply maponpaths.
apply (iso_inv_after_iso (unit_pointwise_iso_from_adj_equivalence H a)).
apply id_right.
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isD).
apply (counit_pointwise_iso_from_adj_equivalence H).
* apply (isotoid _ isD).
apply (counit_pointwise_iso_from_adj_equivalence H).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : D × D, D ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with.
assert (XR := nat_trans_ax (counit_from_are_adjoints (pr2 (pr1 H)))).
cbn in XR. rewrite XR. clear XR.
rewrite assoc.
etrans. apply maponpaths_2.
apply (iso_after_iso_inv (counit_pointwise_iso_from_adj_equivalence H _)).
apply id_left.
Defined.
Definition iso_ob {C D : precategory} (hsD : has_homsets D)
{F G : functor C D} (a : iso (C:= [C, D, hsD]) F G)
: ∏ c, iso (F c) (G c).
Proof.
intro c.
use isopair.
- cbn. apply ((pr1 a : nat_trans _ _ ) c).
- apply is_functor_iso_pointwise_if_iso. apply (pr2 a).
Defined.
Definition isweq_equivalence_on_mor_total {C D : precategory}
(isC : is_univalent C) (isD : is_univalent D)
(F : functor C D) (G : functor D C)
(eta : iso (C:= [_ , _ , pr2 isC ]) (functor_identity C) (F ∙ G))
(eps : iso (C:= [_ , _ , pr2 isD ]) (G ∙ F) (functor_identity D))
: isweq (functor_on_mor_total F).
Proof.
use (gradth _ _ _ _ ).
- apply (functor_on_mor_total G).
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (iso_ob (pr2 isC) eta).
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (iso_ob (pr2 isC) eta).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : C × C, C ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with. rewrite id_right.
rewrite assoc. assert (XR := nat_trans_ax (pr1 eta)).
cbn in XR. rewrite <- XR.
rewrite <- assoc.
rewrite id_right.
etrans. apply maponpaths.
apply (nat_trans_inv_pointwise_inv_after _ _ (pr2 isC) _ _ (pr1 eta)).
apply id_right.
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isD).
apply (iso_ob (pr2 isD) eps).
* apply (isotoid _ isD).
apply (iso_ob (pr2 isD) eps).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : D × D, D ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with.
assert (XR := nat_trans_ax (pr1 eps)).
cbn in XR. rewrite XR. clear XR.
rewrite assoc.
rewrite id_right.
etrans. apply maponpaths_2.
apply (nat_trans_inv_pointwise_inv_before _ _ (pr2 isD) _ _ (pr1 eps)).
apply id_left.
Defined.
:= ∑ (ab : C × C), C⟦pr2 ab, pr1 ab⟧.
Definition morphism_as_total {C : precategory} {a b : C} (f : a --> b) : mor_total C.
Proof.
exists (b,,a).
exact f.
Defined.
Definition source {C} (X : mor_total C) : C := pr2 (pr1 X).
Definition target {C} (X : mor_total C) : C := pr1 (pr1 X).
Definition morphism_from_total {C} (X : mor_total C)
: C ⟦source X, target X⟧
:= pr2 X.
Coercion morphism_from_total : mor_total >-> precategory_morphisms.
Definition functor_on_mor_total {C D : precategory} (F : functor C D)
(p : mor_total C) : mor_total D.
Proof.
exists (F (pr1 (pr1 p)) ,, F (pr2 (pr1 p)) ).
exact (#F p).
Defined.
Definition isweq_left_adj_equivalence_on_mor_total {C D : precategory} (F : functor C D)
(isC : is_univalent C) (isD : is_univalent D)
(H : adj_equivalence_of_precats F)
: isweq (functor_on_mor_total F).
Proof.
use (gradth _ _ _ _ ).
- apply (functor_on_mor_total (adj_equivalence_inv H)).
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (unit_pointwise_iso_from_adj_equivalence H).
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (unit_pointwise_iso_from_adj_equivalence H).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : C × C, C ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with. rewrite id_right.
rewrite assoc. assert (XR := nat_trans_ax (unit_from_are_adjoints (pr2 (pr1 H)))).
cbn in XR. rewrite <- XR.
rewrite <- assoc.
etrans. apply maponpaths.
apply (iso_inv_after_iso (unit_pointwise_iso_from_adj_equivalence H a)).
apply id_right.
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isD).
apply (counit_pointwise_iso_from_adj_equivalence H).
* apply (isotoid _ isD).
apply (counit_pointwise_iso_from_adj_equivalence H).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : D × D, D ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with.
assert (XR := nat_trans_ax (counit_from_are_adjoints (pr2 (pr1 H)))).
cbn in XR. rewrite XR. clear XR.
rewrite assoc.
etrans. apply maponpaths_2.
apply (iso_after_iso_inv (counit_pointwise_iso_from_adj_equivalence H _)).
apply id_left.
Defined.
Definition iso_ob {C D : precategory} (hsD : has_homsets D)
{F G : functor C D} (a : iso (C:= [C, D, hsD]) F G)
: ∏ c, iso (F c) (G c).
Proof.
intro c.
use isopair.
- cbn. apply ((pr1 a : nat_trans _ _ ) c).
- apply is_functor_iso_pointwise_if_iso. apply (pr2 a).
Defined.
Definition isweq_equivalence_on_mor_total {C D : precategory}
(isC : is_univalent C) (isD : is_univalent D)
(F : functor C D) (G : functor D C)
(eta : iso (C:= [_ , _ , pr2 isC ]) (functor_identity C) (F ∙ G))
(eps : iso (C:= [_ , _ , pr2 isD ]) (G ∙ F) (functor_identity D))
: isweq (functor_on_mor_total F).
Proof.
use (gradth _ _ _ _ ).
- apply (functor_on_mor_total G).
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (iso_ob (pr2 isC) eta).
* apply (isotoid _ isC).
apply iso_inv_from_iso. apply (iso_ob (pr2 isC) eta).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : C × C, C ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with. rewrite id_right.
rewrite assoc. assert (XR := nat_trans_ax (pr1 eta)).
cbn in XR. rewrite <- XR.
rewrite <- assoc.
rewrite id_right.
etrans. apply maponpaths.
apply (nat_trans_inv_pointwise_inv_after _ _ (pr2 isC) _ _ (pr1 eta)).
apply id_right.
- intro p.
use total2_paths_f.
+ cbn. destruct p as [[a b] f].
apply pathsdirprod; cbn.
* apply (isotoid _ isD).
apply (iso_ob (pr2 isD) eps).
* apply (isotoid _ isD).
apply (iso_ob (pr2 isD) eps).
+ cbn. destruct p as [[a b] f]. cbn in *.
etrans. apply (transportf_pair (λ x : D × D, D ⟦ pr2 x, pr1 x ⟧)).
cbn.
rewrite transportf_isotoid.
rewrite transportf_isotoid'.
cbn. unfold precomp_with.
assert (XR := nat_trans_ax (pr1 eps)).
cbn in XR. rewrite XR. clear XR.
rewrite assoc.
rewrite id_right.
etrans. apply maponpaths_2.
apply (nat_trans_inv_pointwise_inv_before _ _ (pr2 isD) _ _ (pr1 eps)).
apply id_left.
Defined.
Definition ff_on_isos {C D : precategory} (F : functor C D) : UU
:= ∏ c c', isweq (@functor_on_iso _ _ F c c').
Lemma fully_faithful_impl_ff_on_isos {C D : precategory} (F : functor C D)
: fully_faithful F -> ff_on_isos F.
Proof.
intros Fff c c'.
use gradth.
- intro XR. exists (invmap (weqpair _ (Fff _ _ )) XR). cbn.
apply (ff_reflects_is_iso _ _ _ Fff).
assert (XT := homotweqinvweq (weqpair _ (Fff c c' ))).
cbn in *.
apply (transportb (λ i : _ --> _, is_iso i) (XT (pr1 XR) )).
apply XR.
- cbn. intro i. apply eq_iso. cbn.
apply (homotinvweqweq (weqpair _ (Fff _ _ ))).
- cbn. intro i. apply eq_iso. cbn.
apply (homotweqinvweq (weqpair _ (Fff _ _ ))).
Defined.
Coercion left_adj_from_adj_equiv (X Y : precategory) (K : functor X Y)
(HK : adj_equivalence_of_precats K)
: is_left_adjoint K
:= pr1 HK.
Section about_equivalences.
Variables D1 D2 : precategory.
Variable F : functor D1 D2.
Variable GG : adj_equivalence_of_precats 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 cancel_postcomposition. apply iso_after_iso_inv.
apply id_left.
- intro g.
unfold inv. repeat rewrite functor_comp.
match goal with |[|- ?f1 ;; ?f2 ;; ?f3 = _ ] =>
pathvia ((f1 ;; ηinv _ ) ;; (η _ ;; f2) ;; f3) end.
+ repeat rewrite <- assoc. apply maponpaths.
repeat rewrite assoc.
etrans. Focus 2. do 2 apply cancel_postcomposition. eapply pathsinv0, iso_after_iso_inv.
rewrite id_left. apply idpath.
+ assert (XR := nat_trans_ax η). simpl in XR. rewrite <- XR. clear XR.
repeat rewrite <- assoc.
etrans. do 3 apply maponpaths. apply triangle_id_right_ad. rewrite id_right.
rewrite assoc.
etrans. Focus 2. apply id_left.
apply cancel_postcomposition.
etrans. apply cancel_postcomposition. apply functor_on_inv_from_iso.
assert (XR := triangle_id_right_ad (pr2 (pr1 GG))).
simpl in XR.
unfold ηinv. simpl.
match goal with |[|- inv_from_iso ?e ;; inv_from_iso ?f = _ ] =>
assert (XRR := maponpaths pr1 (iso_inv_of_iso_comp _ _ _ _ f e)) end.
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.
exists (F d).
exact (ηinv d).
Defined.
End about_equivalences.
Section eqv_comp.
Context {A B C : precategory}
{hsA : has_homsets A}
{hsB : has_homsets B}
{hsC : has_homsets C}
{F : functor A B}
{F' : functor B C}.
Section adj_comp.
Hypothesis adF : is_left_adjoint F.
Hypothesis adF' : is_left_adjoint F'.
Let η : functor_precategory _ _ hsA ⟦ _ , _ ⟧ := unit_from_left_adjoint adF.
Let η' : functor_precategory _ _ hsB ⟦ _ , _ ⟧ := unit_from_left_adjoint adF'.
Let ε : functor_precategory _ _ hsB ⟦ _ , _ ⟧ := counit_from_left_adjoint adF.
Let ε' : functor_precategory _ _ hsC ⟦ _ , _ ⟧ := counit_from_left_adjoint adF'.
Let G := right_adjoint adF.
Let G' := right_adjoint adF'.
Definition unit_comp : (functor_precategory A A hsA)
⟦ functor_identity A,
functor_composite (functor_composite F F') (functor_composite G' G) ⟧
:=
let Fη' := # (pre_composition_functor _ _ _ hsB hsB F) η' in
let Fη'G := # (post_composition_functor _ _ _ _ hsA G) Fη' in
(η ;; Fη'G).
Definition counit_comp : (functor_precategory C C hsC)
⟦functor_composite (functor_composite G' G) (functor_composite F F'),
functor_identity C⟧
:=
let G'ε := # (pre_composition_functor _ _ _ hsB hsB G') ε in
let G'εF' := # (post_composition_functor _ _ _ _ hsC F') G'ε in
(G'εF' ;; ε').
Lemma form_adjunction_comp
: form_adjunction
(functor_composite F F') (functor_composite G' G)
unit_comp counit_comp.
Proof.
mkpair; cbn in *; simpl in *.
- intro a.
assert (T1 := triangle_id_left_ad (pr2 adF) a).
assert (T1' := triangle_id_left_ad (pr2 adF') (F a)).
etrans.
+ cbn. etrans. apply assoc.
apply maponpaths_2.
etrans. apply @pathsinv0, functor_comp.
apply maponpaths.
etrans. apply maponpaths_2, functor_comp.
etrans. apply @pathsinv0, assoc.
apply maponpaths, (nat_trans_ax ε).
+ cbn. etrans.
* apply maponpaths_2, maponpaths.
etrans. apply assoc.
etrans. apply maponpaths_2, T1.
apply id_left.
* apply T1'.
- intro c.
assert (T2 := triangle_id_right_ad (pr2 adF) (G' c)).
assert (T2' := triangle_id_right_ad (pr2 adF') c).
etrans.
+ cbn. etrans. apply @pathsinv0, assoc.
apply maponpaths.
etrans. apply @pathsinv0, functor_comp.
apply maponpaths.
etrans. apply maponpaths, functor_comp.
etrans. apply assoc.
apply maponpaths_2, @pathsinv0, (nat_trans_ax η').
+ cbn. etrans.
* apply maponpaths, maponpaths.
etrans. apply @pathsinv0, assoc.
etrans. apply maponpaths, T2'.
apply id_right.
* apply T2.
Qed.
Definition comp_adjunction : is_left_adjoint (functor_composite F F').
Proof.
exists (functor_composite G' G).
exists (unit_comp ,, counit_comp).
apply form_adjunction_comp.
Defined.
End adj_comp.
Hypothesis HF : adj_equivalence_of_precats F.
Hypothesis HF' : adj_equivalence_of_precats F'.
Definition comp_adj_equivalence_of_precats
: adj_equivalence_of_precats (functor_composite F F').
Proof.
exists (comp_adjunction HF HF').
mkpair.
- intro. apply is_iso_comp_is_iso.
+ apply (pr1 (pr2 HF)).
+ simpl. apply functor_is_iso_is_iso, (pr1 (pr2 HF')).
- intro. apply is_iso_comp_is_iso.
+ apply functor_is_iso_is_iso, (pr2 (pr2 HF)).
+ apply (pr2 (pr2 HF')).
Defined.
End eqv_comp.
Section eqv_inv.
Context {A B : precategory}
{hsA : has_homsets A}
{hsB : has_homsets B}
{F : functor A B}.
Hypothesis adEquivF : adj_equivalence_of_precats F.
Let η : functor_precategory _ _ hsA ⟦ _ , _ ⟧ := unit_from_left_adjoint adEquivF.
Let ε : functor_precategory _ _ hsB ⟦ _ , _ ⟧ := counit_from_left_adjoint adEquivF.
Let G := right_adjoint adEquivF.
Let ηiso := unit_iso_from_adj_equivalence_of_precats hsA adEquivF.
Let εiso := counit_iso_from_adj_equivalence_of_precats hsB adEquivF.
Lemma form_adjunction_inv :
form_adjunction G F (inv_from_iso εiso) (inv_from_iso ηiso).
Proof.
split.
- intro b.
apply (pre_comp_with_iso_is_inj _ _ _ _ (#G ((pr1 εiso : nat_trans _ _ ) b))).
+ apply (functor_is_iso_is_iso G).
apply is_functor_iso_pointwise_if_iso. apply pr2.
+ etrans. apply assoc.
etrans. apply maponpaths_2. eapply pathsinv0. apply functor_comp.
etrans. apply maponpaths_2. apply maponpaths.
apply (nat_trans_eq_pointwise (iso_inv_after_iso εiso)).
etrans. apply maponpaths_2. apply functor_id.
etrans. apply id_left.
apply (pre_comp_with_iso_is_inj _ _ _ _ (((pr1 ηiso : nat_trans _ _ ) (G b)))).
* apply is_functor_iso_pointwise_if_iso. apply pr2.
* etrans. apply (nat_trans_eq_pointwise (iso_inv_after_iso ηiso)).
apply pathsinv0.
etrans. apply assoc.
etrans. apply id_right.
assert (XR := triangle_id_right_ad (pr2 (pr1 adEquivF))).
apply XR.
- intro a.
apply (pre_comp_with_iso_is_inj _ _ _ _ (((pr1 εiso : nat_trans _ _ ) _))).
* apply is_functor_iso_pointwise_if_iso. apply pr2.
* etrans. apply assoc.
etrans. apply maponpaths_2.
apply (nat_trans_eq_pointwise (iso_inv_after_iso εiso)).
etrans. apply id_left.
apply (pre_comp_with_iso_is_inj _ _ _ _ (#F ((pr1 ηiso : nat_trans _ _ ) a))).
{ apply (functor_is_iso_is_iso F).
apply is_functor_iso_pointwise_if_iso. apply pr2.
}
etrans. eapply pathsinv0. apply functor_comp.
etrans. apply maponpaths. apply (nat_trans_eq_pointwise (iso_inv_after_iso ηiso )).
etrans. apply functor_id.
apply pathsinv0.
rewrite id_right.
apply triangle_id_left_ad.
Qed.
Definition is_left_adjoint_inv : is_left_adjoint G.
Proof.
mkpair.
- apply F.
- mkpair.
exists (pr1 (iso_inv_from_iso εiso)).
exact (pr1 (iso_inv_from_iso ηiso)).
apply form_adjunction_inv.
Defined.
Definition adj_equivalence_of_precats_inv
: adj_equivalence_of_precats G.
Proof.
exists is_left_adjoint_inv.
split.
- intro b.
apply (is_functor_iso_pointwise_if_iso _ _ hsB _ _
(iso_inv_from_iso εiso) (pr2 (iso_inv_from_iso εiso))).
- intro a.
apply (is_functor_iso_pointwise_if_iso _ _ hsA _ _
(iso_inv_from_iso ηiso) (pr2 (iso_inv_from_iso ηiso))).
Defined.
End eqv_inv.
Section eqv_from_ess_split_and_ff.
Definition split_ess_surj {A B : precategory}
(F : functor A B)
:= ∏ b : B, ∑ a : A, iso (F a) b.
Context {A B : precategory}
{hsA : has_homsets A}
{hsB : has_homsets B}
{F : functor A B}
(Fff : fully_faithful F)
(Fses : split_ess_surj F).
Let Fweq {a b} f := (weq_from_fully_faithful Fff a b f).
Let Finv {a b} g := (invweq (weq_from_fully_faithful Fff a b) g).
Definition G_ff_split_data : functor_data B A.
Proof.
mkpair.
- intro b. exact (pr1 (Fses b)).
- intros b b' f'; cbn.
apply Finv.
exact (pr2 (Fses b) ;; f' ;; inv_from_iso (pr2 (Fses b'))).
Defined.
Definition G_ff_split_ax : is_functor G_ff_split_data.
Proof.
split.
- intro b. cbn. rewrite id_right. simpl.
apply invmap_eq. cbn.
etrans. apply iso_inv_after_iso.
apply pathsinv0, functor_id.
- intros b b1 b2 f g.
apply invmap_eq; cbn.
rewrite functor_comp.
apply pathsinv0.
etrans. apply maponpaths.
apply (homotweqinvweq (weqpair _ (Fff _ _ ))).
etrans. apply maponpaths_2.
apply (homotweqinvweq (weqpair _ (Fff _ _ ))).
repeat rewrite <- assoc. apply maponpaths. apply maponpaths.
repeat rewrite assoc. apply maponpaths_2.
etrans. apply maponpaths_2. apply iso_after_iso_inv.
apply id_left.
Qed.
Definition G_ff_split : functor _ _ := ( _ ,, G_ff_split_ax).
Definition is_nat_trans_ε_ff_split :
is_nat_trans (functor_composite_data G_ff_split_data F)
(functor_identity_data B) (λ b : B, pr2 (Fses b)).
Proof.
intros b b' g;
etrans; [ apply maponpaths_2 ; use homotweqinvweq |];
repeat rewrite <- assoc;
apply maponpaths;
rewrite iso_after_iso_inv;
apply id_right.
Qed.
Definition ε_ff_split
: nat_trans (functor_composite G_ff_split F) (functor_identity B).
Proof.
mkpair.
- intro b.
exact (pr2 (Fses b)).
- apply is_nat_trans_ε_ff_split.
Defined.
Lemma is_nat_trans_η_ff_split :
is_nat_trans (functor_identity_data A)
(functor_composite_data F G_ff_split_data)
(λ a : A, Finv (inv_from_iso (pr2 (Fses (F ((functor_identity A) a)))))).
Proof.
intros a a' f;
apply (invmaponpathsweq (weqpair _ (Fff _ _ )));
cbn;
rewrite functor_comp;
rewrite functor_comp;
etrans; [ apply maponpaths; use homotweqinvweq |];
apply pathsinv0;
etrans; [ apply maponpaths; use homotweqinvweq |];
etrans; [ apply maponpaths_2; use homotweqinvweq |];
repeat rewrite assoc;
rewrite iso_after_iso_inv;
rewrite id_left ;
apply idpath.
Qed.
Definition η_ff_split : nat_trans (functor_identity A) (functor_composite F G_ff_split).
Proof.
mkpair.
- intro a.
apply Finv.
apply (inv_from_iso (pr2 (Fses _ ))).
- apply is_nat_trans_η_ff_split.
Defined.
Lemma form_adjunction_ff_split
: form_adjunction F G_ff_split η_ff_split ε_ff_split.
simpl. split.
* intro a.
cbn.
etrans. apply maponpaths_2. use homotweqinvweq.
apply iso_after_iso_inv.
* intro b.
cbn.
apply (invmaponpathsweq (weqpair _ (Fff _ _ ))).
cbn.
rewrite functor_comp.
rewrite functor_id.
etrans. apply maponpaths. use homotweqinvweq.
etrans. apply maponpaths_2. use homotweqinvweq.
repeat rewrite assoc.
rewrite iso_after_iso_inv.
rewrite id_left.
apply iso_inv_after_iso.
Qed.
Definition adj_equivalence_of_precats_ff_split : adj_equivalence_of_precats F.
Proof.
mkpair.
- exists G_ff_split.
mkpair.
+ exists η_ff_split.
exact ε_ff_split.
+ apply form_adjunction_ff_split.
- split; cbn.
+ intro a.
use (fully_faithful_reflects_iso_proof _ _ _ Fff _ _ (isopair _ _ )).
apply is_iso_inv_from_iso.
+ intro b. apply pr2.
Defined.
End eqv_from_ess_split_and_ff.
Definition split_full {C D : precategory} (F : functor C D) : UU
:= ∏ c c' (f : F c --> F c'), hfiber (#F) f.
Lemma full_from_split_full {C D : precategory} (F : functor C D)
: split_full F -> full F.
Proof.
intros H c c' f.
apply hinhpr, H.
Qed.
Lemma split_full_from_ff {C D : precategory} (F : functor C D)
: fully_faithful F -> split_full F.
Proof.
intros H c c' f.
exists (fully_faithful_inv_hom H c c' f).
apply (homotweqinvweq (weq_from_fully_faithful _ _ _)).
Qed.
Lemma full_from_ff
{D D' : precategory} (F : functor D D')
: fully_faithful F -> full F.
Proof.
intros. apply full_from_split_full, split_full_from_ff; assumption.
Qed.
Lemma right_adj_equiv_is_full {D1 D2 : precategory}
(F : functor D1 D2) (GG : adj_equivalence_of_precats F)
: full (right_adjoint GG).
Proof.
apply full_from_ff, right_adj_equiv_is_ff.
Qed.
Coercion univalent_category_is_univalent : univalent_category >-> is_univalent.
Definition HSET_univalent_category : univalent_category.
Proof.
exists HSET; split.
- apply is_univalent_HSET.
- apply has_homsets_HSET.
Defined.
Definition functor_univalent_category (C : precategory) (D : univalent_category)
: univalent_category.
Proof.
exists (functor_precategory C D (homset_property D)); split.
- apply is_univalent_functor_category.
- apply functor_category_has_homsets.
Defined.
Definition preShv C := functor_univalent_category C^op HSET_univalent_category.
Notation "'Yo'" := (yoneda _ (homset_property _) : functor _ (preShv _)).
Notation "'Yo^-1'" := (invweq (weqpair _ (yoneda_fully_faithful _ (homset_property _) _ _ ))).
Definition yy {C : precategory} {hsC : has_homsets C}
{F : preShv C} {c : C} : ((F : functor _ _) c : hSet) ≃ _ ⟦ yoneda _ hsC c, F⟧.
Proof.
apply invweq.
apply yoneda_weq.
Defined.
Arguments yy {_ _ _ _}.
Lemma yy_natural {C : precategory} {hsC : has_homsets C}
(F : preShv C) (c : C) (A : (F:functor _ _) c : hSet)
c' (f : C⟦c', c⟧) :
yy (# (F : functor _ _) f A) = # (yoneda _ hsC) f ;; yy A.
Proof.
assert (XTT := is_natural_yoneda_iso_inv _ hsC F _ _ f).
apply (toforallpaths _ _ _ XTT).
Qed.
Lemma yy_comp_nat_trans {C : precategory} {hsC : has_homsets C}
(F F' : preShv C) (p : _ ⟦F, F'⟧)
A (v : (F : functor _ _ ) A : hSet)
: yy (hsC:=hsC) v ;; p = yy ((p : nat_trans _ _ ) _ v).
Proof.
apply nat_trans_eq.
- apply has_homsets_HSET.
- intro c. simpl.
apply funextsec. intro f. cbn.
assert (XR := toforallpaths _ _ _ (nat_trans_ax p _ _ f) v ).
cbn in XR.
apply XR.
Qed.
Lemma transportf_yy {C : precategory} {hsC : has_homsets C}
(F : preShv C) (c c' : C) (A : (F : functor _ _ ) c : hSet)
(e : c = c'):
yy (transportf (fun d => (F : functor _ _ ) d : hSet) e A) =
transportf (fun d => preShv C ⟦ yoneda _ hsC d, F⟧) e (yy A).
Proof.
induction e.
apply idpath.
Defined.
Lemma forall_isotid (A : precategory) (a_is : is_univalent A)
(a a' : A) (P : iso a a' -> UU) :
(∏ e, P (idtoiso e)) → ∏ i, P i.
Proof.
intros H i.
rewrite <- (idtoiso_isotoid _ a_is).
apply H.
Defined.
Lemma transportf_isotoid_functor
(A X : precategory) (H : is_univalent A)
(K : functor A X)
(a a' : A) (p : iso a a') (b : X) (f : K a --> b) :
transportf (fun a0 => K a0 --> b) (isotoid _ H p) f = #K (inv_from_iso p) ;; f.
Proof.
rewrite functor_on_inv_from_iso. simpl. cbn.
unfold precomp_with. rewrite id_right.
generalize p.
apply forall_isotid.
- apply H.
- intro e. induction e.
cbn.
rewrite functor_id.
rewrite id_left.
rewrite isotoid_identity_iso.
apply idpath.
Defined.
Lemma idtoiso_transportf_family_of_morphisms (D : precategory)
(A : UU) (B : A -> UU)
(F : ∏ a, B a -> D)
(d d' : D) (deq : d = d')
(R : ∏ a (b : B a), D⟦ F a b, d⟧)
: transportf (λ x, ∏ a b, D⟦ F a b, x⟧) deq R
=
λ a b, R a b ;; idtoiso deq.
Proof.
destruct deq.
apply funextsec.
intro. apply funextsec. intro.
apply pathsinv0.
apply id_right.
Qed.
Lemma idtoiso_concat_pr (C : precategory) (a a' a'' : ob C)
(p : a = a') (q : a' = a'') :
idtoiso p ;; idtoiso q = idtoiso (p @ q).
Proof.
apply pathsinv0.
apply (base_paths _ _ (idtoiso_concat _ _ _ _ _ _ )).
Defined.
Lemma idtoiso_eq_idpath (C : precategory) (a : C) (e : a = a)
(H : e = idpath _ )
: identity a = idtoiso e.
Proof.
rewrite H.
apply idpath.
Qed.
Arguments cancel_postcomposition [C a b c] f f' g _ .
Lemma idtoiso_postcompose_idtoiso_pre {C : precategory} {a b c : C}
(g : a --> b) (f : a --> c)
(p : b = c)
: g = f ;; idtoiso (!p) -> g ;; idtoiso p = f.
Proof.
induction p. simpl.
rewrite id_right.
induction 1.
apply id_right.
Qed.
Lemma postwhisker_isotoid {D : precategory} (H : is_univalent D)
{a b b' : D} (f : a --> b) (p : iso b b')
: transportf (fun b0 => a --> b0) (isotoid _ H p) f
= f ;; p.
Proof.
rewrite <- idtoiso_postcompose.
apply maponpaths, maponpaths, idtoiso_isotoid.
Qed.
Local Notation "[ C , D , hs ]" := (functor_precategory C D hs).
Lemma inv_from_iso_iso_from_fully_faithful_reflection {C D : precategory}
(F : functor C D) (HF : fully_faithful F) (a b : C) (i : iso (F a) (F b))
: inv_from_iso
(iso_from_fully_faithful_reflection HF i) =
iso_from_fully_faithful_reflection HF (iso_inv_from_iso i).
Proof.
cbn.
unfold precomp_with.
apply id_right.
Defined.
Definition nat_iso_from_pointwise_iso (D E : precategory)
(hsE : has_homsets E)
(F G : [D, E, hsE])
(a : ∏ d, iso ((F : functor _ _) d) ((G : functor _ _) d))
(H : is_nat_trans _ _ a)
: iso F G.
Proof.
use functor_iso_from_pointwise_iso .
mkpair.
- intro d. apply a.
- apply H.
- intro d. apply (pr2 (a d)).
Defined.
Lemma iso_from_iso_with_postcomp (D E E' : precategory) hsE hsE'
(F G : functor D E) (H : functor E E')
(Hff : fully_faithful H) :
iso (C:=[D, E', hsE']) (functor_composite F H) (functor_composite G H)
->
iso (C:=[D, E, hsE]) F G.
Proof.
intro a.
use nat_iso_from_pointwise_iso.
- intro d. simpl.
apply (iso_from_fully_faithful_reflection Hff).
apply (functor_iso_pointwise_if_iso _ _ _ _ _ a (pr2 a)).
- abstract (
simpl; intros d d' f;
assert (XR := nat_trans_ax (pr1 a : nat_trans _ _ ));
simpl in XR;
apply (invmaponpathsweq (weq_from_fully_faithful Hff _ _ ));
simpl; rewrite functor_comp; rewrite functor_comp;
assert (XTT:=homotweqinvweq (weq_from_fully_faithful Hff (F d') (G d') ));
simpl in *;
etrans; [apply maponpaths; apply XTT |];
clear XTT;
assert (XTT:=homotweqinvweq (weq_from_fully_faithful Hff (F d) (G d) ));
simpl in *;
etrans; [| apply cancel_postcomposition; apply (!XTT _ )];
apply XR
).
Defined.
Definition functor_assoc_iso (D1 D2 D3 D4 : precategory) hsD4
(F : functor D1 D2) (G : functor D2 D3) (H : functor D3 D4) :
iso (C:=[D1,D4,hsD4])
(functor_composite (functor_composite F G) H)
(functor_composite F (functor_composite G H)).
Proof.
use nat_iso_from_pointwise_iso.
- intro d. apply identity_iso.
- abstract (
intros x x' f;
rewrite id_left;
rewrite id_right;
apply idpath
).
Defined.
Definition functor_comp_id_iso (D1 D2 : precategory) hsD2
(F : functor D1 D2) :
iso (C:=[D1,D2,hsD2]) (functor_composite F (functor_identity _ )) F.
Proof.
use nat_iso_from_pointwise_iso.
- intro. apply identity_iso.
- abstract (
intros x x' f;
rewrite id_left;
rewrite id_right;
apply idpath
).
Defined.
Definition functor_precomp_iso (D1 D2 D3 : precategory) hsD3
(F : functor D1 D2) (G H : functor D2 D3) :
iso (C:=[D2,D3,hsD3]) G H ->
iso (C:=[D1,D3,hsD3]) (functor_composite F G)
(functor_composite F H).
Proof.
intro a.
use nat_iso_from_pointwise_iso.
- intro d. apply (functor_iso_pointwise_if_iso _ _ _ _ _ a (pr2 a)).
- abstract (intros x x' f; apply (nat_trans_ax (pr1 a))).
Defined.
Definition adj_from_equiv (D1 D2 : precategory) (F : functor D1 D2):
adj_equivalence_of_precats F → is_left_adjoint F := fun x => pr1 x.
Coercion adj_from_equiv : adj_equivalence_of_precats >-> is_left_adjoint.
Definition reflects_pullbacks {C D : precategory} (F : functor C D) : UU
:= ∏ {a b c d : C}{f : C ⟦b, a⟧} {g : C ⟦c, a⟧} {h : C⟦d, b⟧} {k : C⟦d,c⟧}
(H : h · f = k · g),
isPullback _ _ _ _ (functor_on_square _ _ F H) -> isPullback _ _ _ _ H.
Lemma ff_reflects_pullbacks {C D : category} {F : functor C D}
(F_ff : fully_faithful F) : reflects_pullbacks F.
Proof.
do 10 intro.
use (isPullback_preimage_square _ _ _ _ _ _ X).
- apply homset_property.
- apply F_ff.
Defined.
Section Square_Transfers.
Definition commutes_and_is_pullback {C : precategory} {a b c d : C}
(f : b --> a) (g : c --> a) (p1 : d --> b) (p2 : d --> c)
: UU
:= ∑ (H : p1 ;; f = p2 ;; g), isPullback _ _ _ _ H.
Lemma commuting_square_transfer_iso {C : precategory}
{a b c d : C}
{f : b --> a} {g : c --> a} {p1 : d --> b} {p2 : d --> c}
{a' b' c' d' : C}
{f' : b' --> a'} {g' : c' --> a'} {p1' : d' --> b'} {p2' : d' --> c'}
{i_a : a --> a'} {i_b : b --> b'} {i_c : c --> c'} {i_d : iso d d'}
(i_f : f ;; i_a = i_b ;; f') (i_g : g ;; i_a = i_c ;; g')
(i_p1 : p1 ;; i_b = i_d ;; p1') (i_p2 : p2 ;; i_c = i_d ;; p2')
: p1 ;; f = p2;; g
-> p1' ;; f' = p2' ;; g'.
Proof.
intro H.
refine (pre_comp_with_iso_is_inj _ _ _ _ i_d _ _ _ _).
exact (pr2 i_d). rewrite 2 assoc.
rewrite <- i_p1, <- i_p2.
rewrite <- 2 assoc.
rewrite <- i_f, <- i_g.
rewrite 2 assoc.
apply maponpaths_2, H.
Qed.
Lemma isPullback_transfer_iso {C : category}
{a b c d : C}
{f : b --> a} {g : c --> a} {p1 : d --> b} {p2 : d --> c}
(H : p1 ;; f = p2;; g)
{a' b' c' d' : C}
{f' : b' --> a'} {g' : c' --> a'} {p1' : d' --> b'} {p2' : d' --> c'}
(H' : p1' ;; f' = p2' ;; g')
{i_a : iso a a'} {i_b : iso b b'} {i_c : iso c c'} {i_d : iso d d'}
(i_f : f ;; i_a = i_b ;; f') (i_g : g ;; i_a = i_c ;; g')
(i_p1 : p1 ;; i_b = i_d ;; p1') (i_p2 : p2 ;; i_c = i_d ;; p2')
: isPullback _ _ _ _ H
-> isPullback _ _ _ _ H'.
Proof.
intros Hpb.
apply (mk_isPullback _ ).
intros X h k H''.
simple refine (tpair _ _ _ ).
- simple refine (tpair _ _ _ ).
{ refine ( _ ;; i_d ).
simple refine (PullbackArrow (mk_Pullback _ _ _ _ _ _ Hpb) _ _ _ _).
+ exact (h ;; iso_inv_from_iso i_b).
+ exact (k ;; iso_inv_from_iso i_c).
+ abstract (
apply (post_comp_with_iso_is_inj _ _ _ i_a (pr2 _));
repeat rewrite <- assoc;
rewrite i_f, i_g;
eapply @pathscomp0;
[ apply maponpaths; rewrite assoc;
apply maponpaths_2, iso_after_iso_inv
| eapply @pathsinv0, @pathscomp0;
[ apply maponpaths; rewrite assoc;
apply maponpaths_2, iso_after_iso_inv
| rewrite 2 id_left; apply @pathsinv0, H'']] ).
}
cbn; split;
abstract (
rewrite <- assoc;
eapply @pathscomp0;
[ apply maponpaths, @pathsinv0;
try apply i_p2; try apply i_p1
| rewrite assoc;
eapply @pathscomp0;
[ apply maponpaths_2;
try apply (PullbackArrow_PullbackPr2 (mk_Pullback _ _ _ _ _ _ _));
try apply (PullbackArrow_PullbackPr1 (mk_Pullback _ _ _ _ _ _ _))
| rewrite <- assoc, iso_after_iso_inv; apply id_right]] ).
- intros hk'.
apply subtypeEquality.
intro; apply isapropdirprod; apply homset_property.
cbn.
apply (post_comp_with_iso_is_inj _ _ _ (iso_inv_from_iso i_d) (pr2 _)).
eapply @pathscomp0. Focus 2.
rewrite <- assoc. cbn. rewrite iso_inv_after_iso. eapply pathsinv0, id_right.
apply PullbackArrowUnique; cbn.
+ apply (post_comp_with_iso_is_inj _ _ _ i_b (pr2 _)).
repeat rewrite <- assoc.
rewrite i_p1, iso_after_iso_inv, id_right.
eapply @pathscomp0.
apply maponpaths. rewrite assoc, iso_after_iso_inv. apply id_left.
apply (pr1 (pr2 hk')).
+ apply (post_comp_with_iso_is_inj _ _ _ i_c (pr2 _)).
repeat rewrite <- assoc.
rewrite i_p2, iso_after_iso_inv, id_right.
eapply @pathscomp0.
apply maponpaths. rewrite assoc, iso_after_iso_inv. apply id_left.
apply (pr2 (pr2 hk')).
Qed.
Lemma commutes_and_is_pullback_transfer_iso {C : category}
{a b c d : C}
{f : b --> a} {g : c --> a} {p1 : d --> b} {p2 : d --> c}
{a' b' c' d' : C}
{f' : b' --> a'} {g' : c' --> a'} {p1' : d' --> b'} {p2' : d' --> c'}
{i_a : iso a a'} {i_b : iso b b'} {i_c : iso c c'} {i_d : iso d d'}
(i_f : f ;; i_a = i_b ;; f') (i_g : g ;; i_a = i_c ;; g')
(i_p1 : p1 ;; i_b = i_d ;; p1') (i_p2 : p2 ;; i_c = i_d ;; p2')
(H : p1 ;; f = p2 ;; g) (P : isPullback _ _ _ _ H)
: commutes_and_is_pullback f' g' p1' p2'.
Proof.
exists (commuting_square_transfer_iso i_f i_g i_p1 i_p2 H).
exact (isPullback_transfer_iso _ _ i_f i_g i_p1 i_p2 P).
Qed.
End Square_Transfers.
Section on_pullbacks.
Variable C : precategory.
Variable hs : has_homsets C.
Variables a b c d : C.
Variables (f : a --> b) (g : a --> c) (k : b --> d) (h : c --> d).
f
a----b
g | | k
| |
c----d
h
Variable sqr_comm : f ;; k = g ;; h.
Variable Pb : isPullback k h f g sqr_comm.
Lemma square_morphism_equal k' (e : k' = k) : f ;; k' = g ;; h.
Proof.
rewrite e. assumption.
Defined.
Lemma isPb_morphism_equal k' (e : k' = k) :
isPullback k' h f g (square_morphism_equal _ e).
Proof.
match goal with |[|- isPullback _ _ _ _ ?HH] => generalize HH end.
rewrite e.
intro.
apply Pb.
Defined.
Local Definition Pbb : Pullback k h.
Proof.
unshelve refine (mk_Pullback _ _ _ _ _ _ _ ).
- apply a.
- apply f.
- apply g.
- apply sqr_comm.
- apply Pb.
Defined.
Definition map_into_Pb {e : C} (x : e --> b) (y : e --> c)
: x ;; k = y ;; h → e --> a.
Proof.
intro H.
unshelve refine (PullbackArrow Pbb _ _ _ _ ).
- apply x.
- apply y.
- apply H.
Defined.
Definition Pb_map_commutes_1 {e : C} (x : e --> b) (y : e --> c) H
: map_into_Pb x y H ;; f = x.
Proof.
assert (P:=PullbackArrow_PullbackPr1 Pbb).
apply P.
Qed.
Definition Pb_map_commutes_2 {e : C} (x : e --> b) (y : e --> c) H
: map_into_Pb x y H ;; g = y.
Proof.
assert (P:=PullbackArrow_PullbackPr2 Pbb).
apply P.
Qed.
Lemma map_into_Pb_unique (e : C) (x y : e --> a)
: x ;; f = y ;; f → x ;; g = y ;; g → x = y.
Proof.
intros H H'.
set (T:=@map_into_Pb _ (x ;; f)(y ;; g)).
assert (TH : x ;; f ;; k = y ;; g ;; h).
{ rewrite H. repeat rewrite <- assoc. rewrite sqr_comm. apply idpath. }
pathvia (T TH).
apply PullbackArrowUnique. apply idpath. assumption.
apply pathsinv0. apply PullbackArrowUnique. apply pathsinv0; assumption.
apply idpath.
Qed.
Lemma postcomp_pb_with_iso (y : C) (r : y --> d) (i : iso b y) (Hi : i ;; r = k) :
∑ H : f ;; i ;; r = g ;; h, isPullback _ _ _ _ H.
Proof.
simple refine (@commutes_and_is_pullback_transfer_iso (C,,hs)
_ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _
_ Pb);
try apply identity_iso;
try rewrite id_left;
try rewrite id_right;
try apply idpath.
apply pathsinv0, Hi.
Qed.
Lemma isPullback_indepdent_of_path (sqr_comm' : f ;; k = g ;; h)
: isPullback _ _ _ _ (sqr_comm').
Proof.
exact Pb.
Defined.
Lemma is_symmetric'_isPullback
: isPullback _ _ _ _ (!sqr_comm) -> isPullback _ _ _ _ sqr_comm.
Proof.
refine (is_symmetric_isPullback hs _).
Defined.
End on_pullbacks.
Arguments map_into_Pb {_ _ _ _ _ _ _ _ _ } _ _ {_} _ _ _ .
Arguments map_into_Pb_unique {_ _ _ _ _ _ _ _ _} _ _ {_} _ _ _ _ .
Section Pullbacks_hSet.
Lemma isPullback_HSET {P A B C : HSET}
(p1 : P --> A) (p2 : P --> B) (f : A --> C) (g : B --> C) (ep : p1 ;; f = p2 ;; g)
: (∏ a b (e : f a = g b), ∃! ab, p1 ab = a × p2 ab = b)
-> isPullback _ _ _ _ ep.
Proof.
intros H X h k ehk.
set (H_existence := fun a b e => pr1 (H a b e)).
set (H_uniqueness := fun a b e x x' => base_paths _ _ (proofirrelevancecontr (H a b e) x x')).
apply iscontraprop1.
- apply invproofirrelevance.
intros hk hk'.
apply subtypeEquality. { intro. apply isapropdirprod; apply setproperty. }
destruct hk as [hk [eh ek]], hk' as [hk' [eh' ek']]; simpl.
apply funextsec; intro x.
refine (H_uniqueness (h x) (k x) _ (_,,_) (_,,_)).
apply (toforallpaths _ _ _ ehk).
split. apply (toforallpaths _ _ _ eh). apply (toforallpaths _ _ _ ek).
split. apply (toforallpaths _ _ _ eh'). apply (toforallpaths _ _ _ ek').
- mkpair.
+ intros x. refine (pr1 (H_existence (h x) (k x) _)). apply (toforallpaths _ _ _ ehk).
+ simpl.
split; apply funextsec; intro x.
apply (pr1 (pr2 (H_existence _ _ _))). apply (pr2 (pr2 (H_existence _ _ _))).
Qed.
Lemma pullback_HSET_univprop_elements {P A B C : HSET}
{p1 : HSET ⟦ P, A ⟧} {p2 : HSET ⟦ P, B ⟧}
{f : HSET ⟦ A, C ⟧} {g : HSET ⟦ B, C ⟧}
(ep : p1 ;; f = p2 ;; g)
(pb : isPullback f g p1 p2 ep)
: (∏ a b (e : f a = g b), ∃! ab, p1 ab = a × p2 ab = b).
Proof.
intros a b e.
set (Pb := (mk_Pullback _ _ _ _ _ _ pb)).
apply iscontraprop1.
- apply invproofirrelevance; intros [ab [ea eb]] [ab' [ea' eb']].
apply subtypeEquality; simpl.
intros x; apply isapropdirprod; apply setproperty.
refine (@toforallpaths unitset _ (fun _ => ab) (fun _ => ab') _ tt).
refine (MorphismsIntoPullbackEqual pb _ _ _ _ );
apply funextsec; intros []; cbn;
(eapply @pathscomp0; [ eassumption | apply pathsinv0; eassumption]).
- simple refine (_,,_).
refine (_ tt).
refine (PullbackArrow Pb (unitset : HSET)
(fun _ => a) (fun _ => b) _).
apply funextsec; intro; exact e.
simpl; split.
+ generalize tt; apply toforallpaths.
apply (PullbackArrow_PullbackPr1 Pb unitset).
+ generalize tt; apply toforallpaths.
apply (PullbackArrow_PullbackPr2 Pb unitset).
Defined.
Lemma pullback_HSET_elements_unique {P A B C : HSET}
{p1 : HSET ⟦ P, A ⟧} {p2 : HSET ⟦ P, B ⟧}
{f : HSET ⟦ A, C ⟧} {g : HSET ⟦ B, C ⟧}
{ep : p1 ;; f = p2 ;; g}
(pb : isPullback f g p1 p2 ep)
(ab ab' : P : hSet)
(e1 : p1 ab = p1 ab') (e2 : p2 ab = p2 ab')
: ab = ab'.
Proof.
set (temp := proofirrelevancecontr
(pullback_HSET_univprop_elements _ pb (p1 ab') (p2 ab')
(toforallpaths _ _ _ ep ab'))).
refine (maponpaths pr1 (temp (ab,, _) (ab',, _))).
- split; assumption.
- split; apply idpath.
Qed.
Lemma square_commutes_preShv_to_pointwise {C : precategory} (hsC : has_homsets C)
{X Y Z W : preShv C}
{f : Y --> X} {g : Z --> X} {p1 : W --> Y} {p2 : W --> Z}
(e : p1 ;; f = p2 ;; g)
(c : C)
: ((p1 : nat_trans _ _) c) ;; ((f : nat_trans _ _) c)
= ((p2 : nat_trans _ _) c) ;; ((g : nat_trans _ _) c).
Proof.
apply (nat_trans_eq_pointwise e).
Qed.
Lemma isPullback_preShv_to_pointwise {C : precategory} (hsC : has_homsets C)
{X Y Z W : preShv C}
{f : Y --> X} {g : Z --> X} {p1 : W --> Y} {p2 : W --> Z}
{e : p1 ;; f = p2 ;; g} (pb : isPullback _ _ _ _ e)
(c : C)
: isPullback ((f : nat_trans _ _) c) ((g : nat_trans _ _) c)
((p1 : nat_trans _ _) c) ((p2 : nat_trans _ _) c)
(square_commutes_preShv_to_pointwise hsC e c).
Proof.
set (XR := @isLimFunctor_is_pointwise_Lim C^op HSET has_homsets_HSET
pullback_graph).
set (XT1 := pullback_diagram _ f g).
specialize (XR XT1).
transparent assert
(XH : (∏ a : C^op,
LimCone
(@colimits.diagram_pointwise C^op HSET has_homsets_HSET
pullback_graph XT1 a))).
{ intro. apply LimConeHSET. }
specialize (XR XH).
specialize (XR W).
set (XT := PullbCone _ _ _ _ p1 p2 e).
specialize (XR XT).
transparent assert (XTT : (isLimCone XT1 W XT)).
{ apply @equiv_isPullback_1.
apply functor_category_has_homsets.
assumption.
}
specialize (XR XTT c).
intros S h k H.
specialize (XR S).
simpl in XR.
transparent assert (
HC : (cone
(@colimits.diagram_pointwise C^op HSET has_homsets_HSET
pullback_graph (pullback_diagram (preShv C) f g) c) S)).
{ use mk_cone.
apply three_rec_dep; cbn.
- apply h.
- simpl. apply (h ;; (pr1 f c)).
- apply k.
- use three_rec_dep; use three_rec_dep.
+ exact (Empty_set_rect _ ).
+ intro. apply idpath.
+ exact (Empty_set_rect _ ).
+ exact (Empty_set_rect _ ).
+ exact (Empty_set_rect _ ).
+ exact (Empty_set_rect _ ).
+ exact (Empty_set_rect _ ).
+ intro; apply (!H).
+ exact (Empty_set_rect _ ).
}
specialize (XR HC).
mkpair.
- exists (pr1 (iscontrpr1 XR)).
split.
+ apply (pr2 (pr1 XR) One).
+ apply (pr2 (pr1 XR) Three).
- intro t.
apply subtypeEquality.
+ intro. apply isapropdirprod; apply has_homsets_HSET.
+ simpl.
apply path_to_ctr.
destruct t as [t [H1 H2]].
use three_rec_dep.
* apply H1.
* destruct H1.
apply idpath.
* apply H2.
Qed.
End Pullbacks_hSet.
will be an instance of a general lemma to be proved
in UniMath
Definition isaprop_Pullback (C : precategory) (H : is_univalent C)
(a b c : C) (f : b --> a) (g : c --> a)
: isaprop (Pullback f g).
Proof.
unfold Pullback.
apply invproofirrelevance.
unfold Pullback.
intros Pb Pb'.
apply subtypeEquality.
- intro; apply isofhleveltotal2.
+ destruct H as [H1 H2]. apply H2.
+ intros; apply isaprop_isPullback.
- apply (total2_paths_f (isotoid _ H (iso_from_Pullback_to_Pullback Pb Pb' ))).
rewrite transportf_dirprod, transportf_isotoid.
rewrite inv_from_iso_iso_from_Pullback.
rewrite transportf_isotoid.
rewrite inv_from_iso_iso_from_Pullback.
destruct Pb as [Cone bla];
destruct Pb' as [Cone' bla'];
simpl in *.
destruct Cone as [p [h k]];
destruct Cone' as [p' [h' k']];
simpl in *.
unfold from_Pullback_to_Pullback;
rewrite PullbackArrow_PullbackPr2, PullbackArrow_PullbackPr1.
apply idpath.
Qed.
Section Pullback_Unique_Up_To_Iso.
Variable CC : precategory.
Variables A B C D A' B' : CC.
Variables (f : A --> B) (g : A --> C) (k : B --> D) (j : C --> D) (H : f ;; k = g ;; j)
(pb : isPullback _ _ _ _ H).
Variables (f' : A' --> B') (g' : A' --> C) (r : B' --> D) (h : iso B B').
Variable (H' : f' ;; r = g' ;; j).
Variable (pb' : isPullback _ _ _ _ H').
Variable (T : h ;; r = k).
Definition map_to_2nd_pb : A --> A'.
Proof.
unshelve refine (map_into_Pb H' pb' _ _ _ ).
- exact (f ;; h).
- exact g.
- eapply pathscomp0. Focus 2. apply H.
eapply pathscomp0. apply (!assoc _ _ _ ).
apply maponpaths. apply T.
Defined.
Definition map_to_1st_pb : A' --> A.
Proof.
unshelve refine (map_into_Pb H pb _ _ _ ).
- exact (f';; inv_from_iso h).
- exact g'.
- eapply pathscomp0; [| apply H'].
eapply pathscomp0; [ apply (!assoc _ _ _ ) |].
apply maponpaths. apply iso_inv_on_right.
apply (!T).
Defined.
Lemma inv1 : map_to_2nd_pb ;; map_to_1st_pb = identity _ .
Proof.
apply (map_into_Pb_unique H pb).
- rewrite id_left.
rewrite <- assoc.
unfold map_to_1st_pb.
rewrite Pb_map_commutes_1.
rewrite assoc.
apply pathsinv0.
apply iso_inv_on_left.
unfold map_to_2nd_pb.
match goal with |[ |- map_into_Pb ?H1 ?pb1 ?x1 ?y1 ?R1 ;; _ = _ ] => assert
(T1:=@Pb_map_commutes_1 _ _ _ _ _ _ _ _ _ H' pb' _ x1 y1 R1) end.
apply T1.
- rewrite id_left.
rewrite <- assoc.
unfold map_to_1st_pb.
rewrite Pb_map_commutes_2.
unfold map_to_2nd_pb.
apply Pb_map_commutes_2.
Qed.
Lemma inv2 : map_to_1st_pb ;; map_to_2nd_pb = identity _ .
Proof.
apply (map_into_Pb_unique H' pb').
- rewrite id_left.
rewrite <- assoc.
unfold map_to_2nd_pb.
rewrite Pb_map_commutes_1.
rewrite assoc.
unfold map_to_1st_pb.
rewrite Pb_map_commutes_1.
rewrite <- assoc.
rewrite iso_after_iso_inv.
apply id_right.
- rewrite id_left.
rewrite <- assoc.
unfold map_to_2nd_pb.
rewrite Pb_map_commutes_2.
unfold map_to_1st_pb.
apply Pb_map_commutes_2.
Qed.
Definition iso_to_second_pb : iso A A'.
Proof.
exists map_to_2nd_pb.
simple refine (is_iso_qinv _ map_to_1st_pb _ ).
split.
- apply inv1.
- apply inv2.
Defined.
End Pullback_Unique_Up_To_Iso.
Arguments map_into_Pb {_ _ _ _ _} _ _ _ _ _ _ {_} _ _ _ .
Arguments Pb_map_commutes_1 {_ _ _ _ _} _ _ _ _ _ _ {_} _ _ _ .
Arguments Pb_map_commutes_2 {_ _ _ _ _} _ _ _ _ _ _ {_} _ _ _ .
(a b c : C) (f : b --> a) (g : c --> a)
: isaprop (Pullback f g).
Proof.
unfold Pullback.
apply invproofirrelevance.
unfold Pullback.
intros Pb Pb'.
apply subtypeEquality.
- intro; apply isofhleveltotal2.
+ destruct H as [H1 H2]. apply H2.
+ intros; apply isaprop_isPullback.
- apply (total2_paths_f (isotoid _ H (iso_from_Pullback_to_Pullback Pb Pb' ))).
rewrite transportf_dirprod, transportf_isotoid.
rewrite inv_from_iso_iso_from_Pullback.
rewrite transportf_isotoid.
rewrite inv_from_iso_iso_from_Pullback.
destruct Pb as [Cone bla];
destruct Pb' as [Cone' bla'];
simpl in *.
destruct Cone as [p [h k]];
destruct Cone' as [p' [h' k']];
simpl in *.
unfold from_Pullback_to_Pullback;
rewrite PullbackArrow_PullbackPr2, PullbackArrow_PullbackPr1.
apply idpath.
Qed.
Section Pullback_Unique_Up_To_Iso.
Variable CC : precategory.
Variables A B C D A' B' : CC.
Variables (f : A --> B) (g : A --> C) (k : B --> D) (j : C --> D) (H : f ;; k = g ;; j)
(pb : isPullback _ _ _ _ H).
Variables (f' : A' --> B') (g' : A' --> C) (r : B' --> D) (h : iso B B').
Variable (H' : f' ;; r = g' ;; j).
Variable (pb' : isPullback _ _ _ _ H').
Variable (T : h ;; r = k).
Definition map_to_2nd_pb : A --> A'.
Proof.
unshelve refine (map_into_Pb H' pb' _ _ _ ).
- exact (f ;; h).
- exact g.
- eapply pathscomp0. Focus 2. apply H.
eapply pathscomp0. apply (!assoc _ _ _ ).
apply maponpaths. apply T.
Defined.
Definition map_to_1st_pb : A' --> A.
Proof.
unshelve refine (map_into_Pb H pb _ _ _ ).
- exact (f';; inv_from_iso h).
- exact g'.
- eapply pathscomp0; [| apply H'].
eapply pathscomp0; [ apply (!assoc _ _ _ ) |].
apply maponpaths. apply iso_inv_on_right.
apply (!T).
Defined.
Lemma inv1 : map_to_2nd_pb ;; map_to_1st_pb = identity _ .
Proof.
apply (map_into_Pb_unique H pb).
- rewrite id_left.
rewrite <- assoc.
unfold map_to_1st_pb.
rewrite Pb_map_commutes_1.
rewrite assoc.
apply pathsinv0.
apply iso_inv_on_left.
unfold map_to_2nd_pb.
match goal with |[ |- map_into_Pb ?H1 ?pb1 ?x1 ?y1 ?R1 ;; _ = _ ] => assert
(T1:=@Pb_map_commutes_1 _ _ _ _ _ _ _ _ _ H' pb' _ x1 y1 R1) end.
apply T1.
- rewrite id_left.
rewrite <- assoc.
unfold map_to_1st_pb.
rewrite Pb_map_commutes_2.
unfold map_to_2nd_pb.
apply Pb_map_commutes_2.
Qed.
Lemma inv2 : map_to_1st_pb ;; map_to_2nd_pb = identity _ .
Proof.
apply (map_into_Pb_unique H' pb').
- rewrite id_left.
rewrite <- assoc.
unfold map_to_2nd_pb.
rewrite Pb_map_commutes_1.
rewrite assoc.
unfold map_to_1st_pb.
rewrite Pb_map_commutes_1.
rewrite <- assoc.
rewrite iso_after_iso_inv.
apply id_right.
- rewrite id_left.
rewrite <- assoc.
unfold map_to_2nd_pb.
rewrite Pb_map_commutes_2.
unfold map_to_1st_pb.
apply Pb_map_commutes_2.
Qed.
Definition iso_to_second_pb : iso A A'.
Proof.
exists map_to_2nd_pb.
simple refine (is_iso_qinv _ map_to_1st_pb _ ).
split.
- apply inv1.
- apply inv2.
Defined.
End Pullback_Unique_Up_To_Iso.
Arguments map_into_Pb {_ _ _ _ _} _ _ _ _ _ _ {_} _ _ _ .
Arguments Pb_map_commutes_1 {_ _ _ _ _} _ _ _ _ _ _ {_} _ _ _ .
Arguments Pb_map_commutes_2 {_ _ _ _ _} _ _ _ _ _ _ {_} _ _ _ .
Section Unorganised.
Lemma transportf_pshf {C : precategory}
{P P' : preShv C} (e : P = P')
{c : C} (x : (P : functor _ _) c : hSet)
: transportf (fun Q : preShv C => (Q : functor _ _) c : hSet) e x
= ((idtoiso e : _ --> _) : nat_trans _ _) c x.
Proof.
destruct e; apply idpath.
Qed.
Lemma transportf_isotoid_pshf {C : precategory}
{P P' : preShv C} (i : iso P P')
{c : C} (x : (P : functor _ _) c : hSet)
: transportf (fun Q : preShv C => (Q : functor _ _) c : hSet)
(isotoid _ (univalent_category_is_univalent (preShv C)) i) x
= ((i : _ --> _) : nat_trans _ _) c x.
Proof.
etrans. apply transportf_pshf.
refine (toforallpaths _ _ _ _ x).
refine (toforallpaths _ _ _ _ c).
apply maponpaths, maponpaths, idtoiso_isotoid.
Qed.
End Unorganised.