use WeakFunctorCategory to prove GArrow/Reification isomorphism

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

diff --git a/src/ReificationCategory.v b/src/ReificationCategory.v
index d271c02..b4d444a 100644 (file)
--- a/src/ReificationCategory.v
+++ b/src/ReificationCategory.v
@@ -23,7 +23,65 @@ Require Import Coherence_ch7_8.
 Require Import Enrichment_ch2_8.
 Require Import RepresentableStructure_ch7_2.
 Require Import Reification.
+Require Import WeakFunctorCategory.
+Require Import SmallSMMEs.
 
-Definition CategoryOfReifications : Category SMME (fun x y:SMME => Reification (smme_e x) (smme_e y) (mon_i (smme_mon y))).
+Inductive ReificationOrIdentity : SMMEs -> SMMEs -> Type :=
+  | roi_id   : forall smme:SMMEs,                                  ReificationOrIdentity smme smme
+  | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (mon_i s2) -> ReificationOrIdentity s1   s2.
+
+Definition reificationOrIdentityFunc
+  : forall s1 s2, ReificationOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
+  intros.
+  destruct X.
+  exists (fun x => x).
+  apply functor_id.
+  exists (reification_rstar_obj r).
+  apply r.
+  Defined.
+
+Definition compose_reifications (s0 s1 s2:SMMEs) :
+  Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
+  intros.
+  refine
+    {| reification_rstar   := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
+     ; reification_r       := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
+    |}.
+  intro K.
+  set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
+  eapply ni_comp.
+  apply q.
+  clear q.
+  set (reification_commutes X K) as comm1.
+  set (reification_commutes X0 (mon_i s1)) as comm2.
   admit.
-  Qed.
\ No newline at end of file
+  Defined.
+
+Definition reificationOrIdentityComp
+  : forall s1 s2 s3, ReificationOrIdentity s1 s2 -> ReificationOrIdentity s2 s3 -> ReificationOrIdentity s1 s3.
+  intros.
+  destruct X.
+    apply X0.
+  destruct X0.
+    apply (roi_reif _ _ r).
+    apply (roi_reif _ _ (compose_reifications _ _ _ r r0)).
+    Defined.
+
+Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
+  refine {| small_func      := ReificationOrIdentity
+          ; small_func_id   := fun s => roi_id s
+          ; small_func_func := fun smme1 smme2 f => projT2 (reificationOrIdentityFunc _ _ f)
+          ; small_func_comp := reificationOrIdentityComp
+         |}; 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.
+  apply if_id.
+  Defined.
+
+Definition CategoryOfReifications :=
+  WeakFunctorCategory MorphismsOfCategoryOfReifications.