Merge branch 'coq-extraction-baked-in' of /afs/megacz.com/.pub/software/coq-hetmet

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

diff --git a/src/GeneralizedArrowCategory.v b/src/GeneralizedArrowCategory.v
index a9bedbb..745484b 100644 (file)
--- a/src/GeneralizedArrowCategory.v
+++ b/src/GeneralizedArrowCategory.v
@@ -18,40 +18,52 @@ Require Import Enrichment_ch2_8.
 Require Import Subcategories_ch7_1.
 Require Import NaturalTransformations_ch7_4.
 Require Import NaturalIsomorphisms_ch7_5.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
 Require Import MonoidalCategories_ch7_8.
 Require Import Coherence_ch7_8.
 Require Import Enrichment_ch2_8.
+Require Import Enrichments.
 Require Import RepresentableStructure_ch7_2.
 Require Import GeneralizedArrow.
 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
+ *
+ * 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 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 generalizedArrowOrIdentityFobj (s1 s2:SMMEs) (f:GeneralizedArrowOrIdentity s1 s2) : s1 -> s2 :=
+  match f in GeneralizedArrowOrIdentity S1 S2 return S1 -> S2 with
+  | gaoi_id  s       => fun x => x
+  | gaoi_ga s1 s2 f  => fun a => ehom(ECategory:=s2) (enr_c_i (smme_e s2)) (ga_functor_obj f a)
+  end.
 
-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 generalizedArrowOrIdentityFunc s1 s2 (f:GeneralizedArrowOrIdentity s1 s2)
+  : Functor s1 s2 (generalizedArrowOrIdentityFobj _ _ f) :=
+  match f with
+  | gaoi_id  s       => functor_id _
+  | gaoi_ga s1 s2 f  => ga_functor f >>>> HomFunctor s2 (enr_c_i s2)
+  end.
+
+Instance compose_generalizedArrows (s0 s1 s2:SMMEs) 
+  (g01:GeneralizedArrow s0 s1)(g12:GeneralizedArrow s1 s2) : GeneralizedArrow s0 s2 :=
+    { ga_functor_monoidal := g01 >>⊗>> smme_mon s1 >>⊗>> g12 }.
+  apply ga_host_lang_pure.
+  Defined.
 
 Definition generalizedArrowOrIdentityComp
   : forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3.
@@ -66,7 +78,7 @@ Definition generalizedArrowOrIdentityComp
 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_func := fun smme1 smme2 f => generalizedArrowOrIdentityFunc _ _ f
           ; small_func_comp := generalizedArrowOrIdentityComp
          |}; intros; simpl.
   apply if_id.
@@ -78,7 +90,6 @@ Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs.
     apply if_right_identity.
   unfold mf_F.
   idtac.
-  unfold mf_f.
   apply if_associativity.
   Defined.