update to new coq-categories, base ND_Relation on inert sequences

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

diff --git a/src/GeneralizedArrowCategory.v b/src/GeneralizedArrowCategory.v
index cd43282..3a6a74a 100644 (file)
--- a/src/GeneralizedArrowCategory.v
+++ b/src/GeneralizedArrowCategory.v
@@ -9,6 +9,87 @@
 Generalizable All Variables.
 Require Import Preamble.
 Require Import General.
-Require Import NaturalDeduction.
-Require Import Coq.Strings.String.
-Require Import Coq.Lists.List.
+Require Import Categories_ch1_3.
+Require Import Functors_ch1_4.
+Require Import Isomorphisms_ch1_5.
+Require Import ProductCategories_ch1_6_1.
+Require Import OppositeCategories_ch1_6_2.
+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.
+
+(*
+ * 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 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 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 }.
+
+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 => 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.
+  apply if_associativity.
+  Defined.
+
+Definition CategoryOfGeneralizedArrows :=
+  WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.