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Elimination principles for GroupCoeq #2184

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Elimination principles for GroupCoeq #2184

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a05ecf2
constructors and functoriality for GroupCoeq
Alizter Dec 29, 2024
2ef6585
replace ugly change with explicit wildcat term
Alizter Jan 7, 2025
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86 changes: 79 additions & 7 deletions theories/Algebra/Groups/GroupCoeq.v
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Expand Up @@ -4,17 +4,49 @@ Require Import Truncations.Core.
Require Import Algebra.Groups.Group.
Require Import Colimits.Coeq.
Require Import Algebra.Groups.FreeProduct.
Require Import List.Core.

Local Open Scope mc_scope.
Local Open Scope mc_mult_scope.

(** Coequalizers of group homomorphisms *)

Definition GroupCoeq {A B : Group} (f g : A $-> B) : Group.
Proof.
rapply (AmalgamatedFreeProduct (FreeProduct A A) A B).
1,2: apply FreeProduct_rec.
+ exact grp_homo_id.
+ exact grp_homo_id.
+ exact f.
+ exact g.
nrapply (AmalgamatedFreeProduct (FreeProduct A A) A B).
- exact (FreeProduct_rec (Id _) (Id _)).
- exact (FreeProduct_rec f g).
Defined.

Definition groupcoeq_in {A B : Group} {f g : A $-> B}
: B $-> GroupCoeq f g
:= amal_inr.

Definition groupcoeq_glue {A B : Group} {f g : A $-> B}
: groupcoeq_in (f:=f) (g:=g) $o f $== groupcoeq_in $o g.
Proof.
intros x; simpl.
rewrite <- (right_identity (f x)).
rewrite <- (right_identity (g x)).
rhs_V nrapply (amal_glue (freeproduct_inr x)).
symmetry.
nrapply (amal_glue (freeproduct_inl x)).
Defined.

Definition groupcoeq_rec {A B C : Group} (f g : A $-> B)
(h : B $-> C) (p : h $o f $== h $o g)
: GroupCoeq f g $-> C.
Proof.
snrapply AmalgamatedFreeProduct_rec.
- exact (h $o f).
- exact h.
- snrapply freeproduct_ind_homotopy.
+ refine (cat_assoc _ _ _ $@ _ $@ cat_assoc_opp _ _ _).
exact ((_ $@L freeproduct_rec_beta_inl _ _) $@ cat_idr _
$@ (_ $@L freeproduct_rec_beta_inl _ _)^$).
+ refine (cat_assoc _ _ _ $@ _ $@ cat_assoc_opp _ _ _).
exact ((_ $@L freeproduct_rec_beta_inr _ _) $@ (cat_idr _ $@ p)
$@ (_ $@L freeproduct_rec_beta_inr _ _)^$).
Defined.

Definition equiv_groupcoeq_rec `{Funext} {A B C : Group} (f g : GroupHomomorphism A B)
Expand Down Expand Up @@ -55,6 +87,46 @@ Proof.
pose (q1 := p (freeproduct_inl y)).
simpl in q1.
rewrite 2 right_identity in q1.
exact q1^. }
symmetry.
exact q1. }
hnf; intros; apply path_ishprop.
Defined.

Definition groupcoeq_ind_hprop {G H : Group} {f g : G $-> H}
(P : GroupCoeq f g -> Type) `{forall x, IsHProp (P x)}
(c : forall h, P (groupcoeq_in h))
(Hop : forall x y, P x -> P y -> P (x * y))
: forall x, P x.
Proof.
srapply amalgamatedfreeproduct_ind_hprop.
- intros x.
rewrite <- (right_identity x).
refine ((amal_glue (freeproduct_inl x))^ #_).
simpl.
rewrite (right_identity (f x)).
exact (c (f x)).
- exact c.
- exact Hop.
Defined.

Definition groupcoeq_ind_homotopy {G H K : Group} {f g : G $-> H}
{h h' : GroupCoeq f g $-> K}
(r : h $o groupcoeq_in $== h' $o groupcoeq_in)
: h $== h'.
Proof.
rapply (groupcoeq_ind_hprop _ r).
intros x y p q; by nrapply grp_homo_op_agree.
Defined.

Definition functor_groupcoeq
{G H : Group} {f g : G $-> H} {G' H' : Group} {f' g' : G' $-> H'}
(h : G $-> G') (k : H $-> H')
(p : k $o f $== f' $o h) (q : k $o g $== g' $o h)
: GroupCoeq f g $-> GroupCoeq f' g'.
Proof.
refine (groupcoeq_rec f g (groupcoeq_in $o k) _).
refine (cat_assoc _ _ _ $@ _ $@ cat_assoc_opp _ _ _).
refine ((_ $@L p) $@ _ $@ (_ $@L q^$)).
refine (cat_assoc_opp _ _ _ $@ (_ $@R _) $@ cat_assoc _ _ _).
apply groupcoeq_glue.
Defined.
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