use WeakFunctorCategory to prove GArrow/Reification isomorphism

[coq-hetmet.git] / src / GeneralizedArrowCategory.v

diff --git a/src/GeneralizedArrowCategory.v b/src/GeneralizedArrowCategory.v
index 974fdc3..a9bedbb 100644 (file)
--- a/src/GeneralizedArrowCategory.v
+++ b/src/GeneralizedArrowCategory.v
@@ -23,7 +23,64 @@ Require Import Coherence_ch7_8.
 Require Import Enrichment_ch2_8.
 Require Import RepresentableStructure_ch7_2.
 Require Import GeneralizedArrow.
+Require Import WeakFunctorCategory.
+Require Import SmallSMMEs.
 
-Definition CategoryOfGeneralizedArrows : Category SMME (fun x y => @GeneralizedArrow x _ y).
-  admit.
-  Qed.
+Inductive GeneralizedArrowOrIdentity : SMMEs -> SMMEs -> Type :=
+  | gaoi_id   : forall smme:SMMEs,                             GeneralizedArrowOrIdentity smme smme
+  | gaoi_ga : forall s1 s2:SMMEs, GeneralizedArrow s1 s2  -> GeneralizedArrowOrIdentity s1   s2.
+
+Definition generalizedArrowOrIdentityFunc
+  : forall s1 s2, GeneralizedArrowOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
+  intros.
+  destruct X.
+  exists (fun x => x).
+  apply functor_id.
+  eapply existT.
+  apply (g >>>> RepresentableFunctor s2 (mon_i s2)).
+  Defined.
+
+Definition compose_generalizedArrows (s0 s1 s2:SMMEs) :
+  GeneralizedArrow s0 s1 -> GeneralizedArrow s1 s2 -> GeneralizedArrow s0 s2.
+  intro g01.
+  intro g12.
+  refine
+    {| ga_functor          := g01 >>>> RepresentableFunctor s1 (mon_i s1) >>>> g12 |}.
+    apply MonoidalFunctorsCompose.
+    apply MonoidalFunctorsCompose.
+    apply (ga_functor_monoidal g01).
+    apply (me_mf s1).
+    apply (ga_functor_monoidal g12).
+    Defined.
+
+Definition generalizedArrowOrIdentityComp
+  : forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3.
+  intros.
+  destruct X.
+    apply X0.
+  destruct X0.
+    apply (gaoi_ga _ _ g).
+    apply (gaoi_ga _ _ (compose_generalizedArrows _ _ _ g g0)).
+    Defined.
+
+Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs.
+  refine {| small_func      := GeneralizedArrowOrIdentity
+          ; small_func_id   := fun s => gaoi_id s
+          ; small_func_func := fun smme1 smme2 f => projT2 (generalizedArrowOrIdentityFunc _ _ f)
+          ; small_func_comp := generalizedArrowOrIdentityComp
+         |}; intros; simpl.
+  apply if_id.
+  destruct f as [|fobj f]; simpl in *.
+    apply if_inv.
+    apply if_left_identity.
+  destruct g as [|gobj g]; simpl in *.
+    apply if_inv.
+    apply if_right_identity.
+  unfold mf_F.
+  idtac.
+  unfold mf_f.
+  apply if_associativity.
+  Defined.
+
+Definition CategoryOfGeneralizedArrows :=
+  WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.