constructors and functoriality for GroupCoeq

Signed-off-by: Ali Caglayan <alizter@gmail.com>

<!-- ps-id: 4ccb2a6c-3b11-4ba5-9be2-530334d3f40d -->

theories/Algebra/Groups/GroupCoeq.v

@@ -4,17 +4,51 @@ 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.
+    + change (?f1 $o ?f2 $o ?f3 $o ?f4 $== ?f5 $o ?f6 $o ?f7)
+        with (f1 $o f2 $o (f3 $o f4) $== f5 $o (f6 $o f7)).
+      exact ((_ $@L freeproduct_rec_beta_inl _ _) $@ cat_idr _
+        $@ (_ $@L freeproduct_rec_beta_inl _ _)^$).
+    + change (?f1 $o ?f2 $o ?f3 $o ?f4 $== ?f5 $o ?f6 $o ?f7)
+        with (f1 $o f2 $o (f3 $o f4) $== f5 $o (f6 $o f7)).
+      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)
@@ -55,6 +89,46 @@ 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.