X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FReificationCategory.v;h=f7356796e8719f1589aeac65e8c28be7283fdd9d;hp=eb6b5fdec186cdb4b14784eb7b62ea848f6c4ef1;hb=e536cc4194f350ed6de5d465bcf53fda650b3d12;hpb=992203bb4a221ea2f415c0d14bb34d35af2ee637

diff --git a/src/ReificationCategory.v b/src/ReificationCategory.v
index eb6b5fd..f735679 100644
--- a/src/ReificationCategory.v
+++ b/src/ReificationCategory.v
@@ -23,7 +23,91 @@ Require Import Coherence_ch7_8.
 Require Import Enrichment_ch2_8.
 Require Import RepresentableStructure_ch7_2.
 Require Import Reification.
+Require Import WeakFunctorCategory.
+Require Import SmallSMMEs.
 
-Instance CategoryOfReifications : Category SMME Reification.
-  admit.
-  Qed.
\ No newline at end of file
+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 reificationOrIdentityFobj s1 s2 (f:ReificationOrIdentity s1 s2) : s1 -> s2 :=
+  match f with
+  | roi_id   s       => (fun x => x)
+  | roi_reif s1 s2 f => reification_rstar_obj f
+  end.
+
+Definition reificationOrIdentityFunc
+  : forall s1 s2 (f:ReificationOrIdentity s1 s2), Functor (enr_v s1) (enr_v s2) (reificationOrIdentityFobj s1 s2 f).
+  intros.
+  destruct f.
+  apply functor_id.
+  unfold reificationOrIdentityFobj.
+  apply (reification_rstar_f 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.
+  set (RepresentableFunctor s0 K) as a in *.
+  set (reification_rstar_f X) as a' in *.
+  set (reification_rstar_f X0) as x in *.
+  set (reification_r X K) as b in *.
+  set (reification_r X0 (mon_i s1)) as c in *.
+  set (RepresentableFunctor s2 (mon_i s2)) as c' in *.
+  set (RepresentableFunctor s1 (mon_i s1)) as b' in *.
+  apply (ni_comp(F2:=b >>>> (b' >>>> x))).
+  apply (@ni_respects _ _ _ _ _ _ _ _ _ _ b _ b _ (c >>>> c') _ (b' >>>> x)).
+  apply ni_id.
+  apply comm2.
+  eapply ni_comp.
+  eapply ni_inv.
+  apply (ni_associativity b b' x).
+  eapply ni_inv.
+  eapply ni_comp.
+  eapply ni_inv.
+  apply (ni_associativity a a' x).
+  apply (@ni_respects _ _ _ _ _ _ _ _ _ _ (a >>>> a') _ (b >>>> b') _ x _ x).
+  apply ni_inv.
+  apply comm1.
+  apply ni_id.
+  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 => 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.