Library UniMath.CategoryTheory.limits.graphs.eqdiag
- Custom notion of equality between diagrams (eq_diag) over the same graph
- Transports of cones and cocones between equal diagrams.
- Limits/Colimits are the same for equal diagrams.
- C,D be two categories,
- A and B two functors from C to D
- x an object of C
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Categories.
Require Import UniMath.CategoryTheory.functor_categories.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.CategoryTheory.limits.graphs.colimits.
Require Import UniMath.CategoryTheory.limits.graphs.limits.
Local Open Scope cat.
Lemma is_exists_unique {A : UU} {B : A → UU} (H : ∃! a : A, B a) :
B ( pr1 (iscontrpr1 H)).
Proof.
exact(pr2 (pr1 H)).
Qed.
Lemma transport_swap: ∏ {X Y : UU} (P : X → Y → UU) {x x':X} {y y' : Y}
(e : x = x') (e' : y = y') (p : P x y),
transportf (λ a, P _ a) e' (transportf (λ a, P a _) e p) =
transportf (λ a, P a _) e (transportf (λ a, P _ a) e' p) .
Proof.
intros.
induction e.
induction e'.
apply idpath.
Qed.
Lemma transportf2_comp {X : UU} (P : X → X → UU) (x x' : X)
(ex : x = x') (t:P x x) :
transportf (λ y, P y y) ex t = transportf (λ y, P y x') ex
(transportf (λ y, P x y) ex t).
Proof.
now induction ex.
Qed.
Definition eq_diag {C : category} {g : graph} (d d' : diagram g C) :=
∑ (eq_v : ∏ v: vertex g, dob d v = dob d' v), ∏ (v v':vertex g) (f:edge v v'),
transportf (λ obj, C⟦obj, dob d v'⟧) (eq_v v) (dmor d f) =
transportb (λ obj, C⟦_, obj⟧) (eq_v v') (dmor d' f).
Lemma eq_is_eq_diag {C : category} {g : graph} (d d' : diagram g C) :
d = d' → eq_diag d d'.
Proof.
intro e.
induction e.
∃ (λ x, idpath _).
exact (λ x y z, idpath _).
Qed.
Lemma eq_diag_is_eq {C : category} {g : graph} (d d' : diagram g C) :
eq_diag d d' → d = d'.
Proof.
intros [eqv autreq].
use total2_paths_f.
- apply funextfun.
intro v.
apply eqv.
- rewrite (transportf2_comp
(λ x y : vertex g → C, ∏ a b : vertex g, edge a b →
C ⟦ y a, x b ⟧)).
match goal with |- transportf ?Pf ?x1 (transportf ?Pf2 ?s1 ?s2 ) = _ ⇒
set (e := x1);
set (P := Pf);
set (P2 := Pf2);
set (tp2:=transportf P2 s1 s2);
set (trp := transportf P x1 tp2)
end.
change (trp = pr2 d').
unfold trp.
apply funextsec.
intro v; apply funextsec; intro v'.
apply funextsec; intro ed.
specialize (autreq v v' ed).
rewrite <- (pathsinv0inv0 (eqv v)) in autreq.
apply pathsinv0 in autreq.
apply transportf_transpose in autreq.
unfold dmor in autreq.
rewrite autreq.
rewrite pathsinv0inv0.
etrans.
eapply pathsinv0.
apply ( transport_map (P:=P) (Q:=_) (λ x tp, tp v v' ed)).
etrans.
apply (transportf_funextfun (λ x, C⟦ pr1 d' v,x⟧)).
apply maponpaths.
etrans.
eapply pathsinv0.
apply ( transport_map (P:=P2) (Q:=_) (λ x tp, tp v v' ed)).
apply (transportf_funextfun (λ x, C⟦ x,pr1 d v'⟧)).
Qed.
Lemma sym_eq_diag {C : category} {g : graph} (d d' : diagram g C) :
eq_diag d d' → eq_diag d' d.
Proof.
intros eq_d.
set (eq_d1 := pr1 eq_d).
set (eq_d2 := pr2 eq_d).
use tpair.
- intro v.
apply (! (eq_d1 v)).
-
unfold eq_d1.
assert (heqdag:eq_diag d' d).
+ apply eq_is_eq_diag.
apply pathsinv0.
apply eq_diag_is_eq.
assumption.
+
abstract (cbn;
intros v v' f;
specialize (eq_d2 v v' f);
apply pathsinv0;
unfold transportb;
rewrite pathsinv0inv0;
apply (transportf_transpose (P:=(λ obj : C, C ⟦ obj, dob d' v' ⟧)));
assert (eq_d2':=transportf_transpose (P:=(precategory_morphisms (dob d' v)))
_ _ _ (! eq_d2));
rewrite eq_d2';
unfold transportb; rewrite pathsinv0inv0;
apply (transport_swap (λ a b, C⟦b,a⟧))).
Defined.
Lemma eq_diag_mkcocone :
∏ {C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(heq_d: eq_diag d d')
{c : C} (cc:cocone d c),
cocone d' c.
Proof.
clear.
intros.
destruct heq_d as [heq heq2].
use mk_cocone.
intro v.
use (transportf (λ obj, C⟦obj,_⟧ ) (heq v)); simpl.
apply (coconeIn cc).
abstract(
intros u v e; simpl;
rewrite <- ( coconeInCommutes cc u v e);
apply (pathscomp0 (b:=transportb (precategory_morphisms (dob d' u)) (heq v)
(dmor d' e) · (coconeIn cc v)));
[
unfold transportb; (set (z:= ! heq v));
rewrite <- (pathsinv0inv0 (heq v));
apply pathsinv0;
apply transport_compose|];
etrans; [
apply cancel_postcomposition;
eapply pathsinv0; apply heq2|];
clear;
now destruct (heq u)).
Defined.
Lemma eq_diag_mkcone :
∏ {C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(heq_d: eq_diag d d')
{c : C} (cc:cone d c),
cone d' c.
Proof.
clear.
intros.
set (heq := pr1 heq_d).
set (heq2 := pr2 heq_d).
use mk_cone.
intro v.
apply (transportf (λ obj, C⟦_,obj⟧ ) (heq v) (coneOut cc v)).
abstract(
intros u v e; simpl;
rewrite <- ( coneOutCommutes cc u v e);
etrans;[
apply transport_compose|];
rewrite transport_target_postcompose;
apply cancel_precomposition;
apply transportf_transpose;
etrans;[
apply (transport_swap (λ a b, C⟦a,b⟧))|];
etrans;[
apply maponpaths;
eapply pathsinv0;
apply heq2|];
unfold heq;
induction (pr1 heq_d u);
apply idpath).
Defined.
Lemma eq_diag_islimcone:
∏ {C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
{c : C} {cc:cone d c}
(islimcone : isLimCone _ _ cc) ,
isLimCone _ _ (eq_diag_mkcone d' eq_d cc).
Proof.
intros.
set (eq_d1 := pr1 eq_d);
set (eq_d2 := pr1 eq_d).
set (eq_d' := sym_eq_diag _ _ eq_d).
set (eq_d1' := pr1 eq_d').
set (eq_d2' := pr2 eq_d').
red.
intros c' cc'.
set (cc'2 := eq_diag_mkcone _ eq_d' cc').
specialize (islimcone c' cc'2).
apply (unique_exists (pr1 (pr1 islimcone))).
- intro v.
assert (islim := is_exists_unique islimcone v).
cbn in islim.
cbn.
etrans.
eapply pathsinv0.
apply transport_target_postcompose.
etrans.
apply maponpaths.
apply islim.
apply transportfbinv.
- intro y.
apply impred_isaprop.
intro t.
apply homset_property.
- intros y hy.
apply (path_to_ctr _ _ islimcone).
intro v; specialize (hy v).
cbn.
apply transportf_transpose.
rewrite <- hy.
etrans.
unfold transportb.
rewrite pathsinv0inv0.
apply transport_target_postcompose.
apply idpath.
Qed.
The dual proof .
This proof could be deduced from the previous if there was a lemma
stating that colimits are limits in the dual category.
Lemma eq_diag_iscolimcocone:
∏ {C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
{c : C} {cc:cocone d c}
(islimcone : isColimCocone _ _ cc) ,
isColimCocone _ _ (eq_diag_mkcocone d' eq_d cc).
Proof.
intros.
destruct eq_d as [eq_d1 eq_d2].
set (eq_d := eq_d1,,eq_d2).
set (eq_d'' := sym_eq_diag _ _ eq_d).
set (eq_d1' := pr1 eq_d'').
set (eq_d2' := pr2 eq_d'').
set (eq_d' := (eq_d1',,eq_d2'):eq_diag d' d).
red.
intros c' cc'.
set (cc'2 := eq_diag_mkcocone _ eq_d' cc').
specialize (islimcone c' cc'2).
apply (unique_exists (pr1 (pr1 islimcone))).
- intro v.
assert (islim := is_exists_unique islimcone v).
cbn in islim.
cbn.
etrans.
rewrite <- (pathsinv0inv0 (eq_d1 v)).
eapply pathsinv0.
apply transport_source_precompose.
etrans.
apply maponpaths.
apply islim.
cbn.
now apply (transportbfinv ( (λ x' : C, C ⟦ x', c' ⟧) )).
- intro y.
apply impred_isaprop.
intro t.
apply homset_property.
- intros y hy.
apply (path_to_ctr _ _ islimcone).
intro v; specialize (hy v).
revert hy.
cbn.
intro hy.
apply (transportf_transpose (P:=(λ obj : C, C ⟦ obj, c' ⟧))).
etrans.
apply transport_source_precompose.
unfold transportb.
rewrite pathsinv0inv0.
apply hy.
Qed.
Definition eq_diag_liftcolimcocone
{C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
(cc:ColimCocone d ) : ColimCocone d'
:= mk_ColimCocone _ _ _ (eq_diag_iscolimcocone _ eq_d
(isColimCocone_ColimCocone cc)).
Definition eq_diag_liftlimcone
{C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
(cc:LimCone d ) : LimCone d'
:= mk_LimCone _ _ _ (eq_diag_islimcone _ eq_d
(isLimCone_LimCone cc)).
∏ {C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
{c : C} {cc:cocone d c}
(islimcone : isColimCocone _ _ cc) ,
isColimCocone _ _ (eq_diag_mkcocone d' eq_d cc).
Proof.
intros.
destruct eq_d as [eq_d1 eq_d2].
set (eq_d := eq_d1,,eq_d2).
set (eq_d'' := sym_eq_diag _ _ eq_d).
set (eq_d1' := pr1 eq_d'').
set (eq_d2' := pr2 eq_d'').
set (eq_d' := (eq_d1',,eq_d2'):eq_diag d' d).
red.
intros c' cc'.
set (cc'2 := eq_diag_mkcocone _ eq_d' cc').
specialize (islimcone c' cc'2).
apply (unique_exists (pr1 (pr1 islimcone))).
- intro v.
assert (islim := is_exists_unique islimcone v).
cbn in islim.
cbn.
etrans.
rewrite <- (pathsinv0inv0 (eq_d1 v)).
eapply pathsinv0.
apply transport_source_precompose.
etrans.
apply maponpaths.
apply islim.
cbn.
now apply (transportbfinv ( (λ x' : C, C ⟦ x', c' ⟧) )).
- intro y.
apply impred_isaprop.
intro t.
apply homset_property.
- intros y hy.
apply (path_to_ctr _ _ islimcone).
intro v; specialize (hy v).
revert hy.
cbn.
intro hy.
apply (transportf_transpose (P:=(λ obj : C, C ⟦ obj, c' ⟧))).
etrans.
apply transport_source_precompose.
unfold transportb.
rewrite pathsinv0inv0.
apply hy.
Qed.
Definition eq_diag_liftcolimcocone
{C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
(cc:ColimCocone d ) : ColimCocone d'
:= mk_ColimCocone _ _ _ (eq_diag_iscolimcocone _ eq_d
(isColimCocone_ColimCocone cc)).
Definition eq_diag_liftlimcone
{C : category} {g : graph} {d : diagram g C}
(d' : diagram g C)
(eq_d : eq_diag d d')
(cc:LimCone d ) : LimCone d'
:= mk_LimCone _ _ _ (eq_diag_islimcone _ eq_d
(isLimCone_LimCone cc)).