remove notations from Preamble that come from coq-categories

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

diff --git a/src/ReificationCategory.v b/src/ReificationCategory.v
index f735679..e57df4d 100644 (file)
--- a/src/ReificationCategory.v
+++ b/src/ReificationCategory.v
@@ -26,6 +26,22 @@ Require Import Reification.
 Require Import WeakFunctorCategory.
 Require Import SmallSMMEs.
 
+(*
+ * Technically reifications form merely a *semicategory* (no identity
+ * maps), but one can always freely adjoin identity maps (and nothing
+ * else) to a semicategory to get a category whose non-identity-map
+ * portion is identical to the original semicategory (closing under
+ * composition after putting in the identity maps never produces any
+ * additional maps)
+ *
+ * Also, technically this category has ALL enrichments (not just the
+ * surjective monic monoidal ones), though there maps OUT OF only the
+ * surjective enrichments and INTO only the monic monoidal
+ * enrichments.  It's a big pain to do this in Coq, but sort of might
+ * matter in real life: a language with really severe substructural
+ * restrictions might fail to be monoidally enriched, meaning we can't
+ * use it as a host language.  But that's for the next paper...
+ *)
 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.
@@ -49,23 +65,24 @@ 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_rstar_f := reification_rstar_f X >>>> reification_rstar_f X0
+     ; 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.
+  set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (HomFunctor 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 (HomFunctor 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 *.
+  set (HomFunctor s2 (mon_i s2)) as c' in *.
+  set (HomFunctor 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.