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

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

diff --git a/src/ReificationCategory.v b/src/ReificationCategory.v
index eb6b5fd..9a7c3fd 100644 (file)
--- a/src/ReificationCategory.v
+++ b/src/ReificationCategory.v
@@ -18,12 +18,112 @@ Require Import Enrichment_ch2_8.
 Require Import Subcategories_ch7_1.
 Require Import NaturalTransformations_ch7_4.
 Require Import NaturalIsomorphisms_ch7_5.
+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 Reification.
+Require Import WeakFunctorCategory.
 
-Instance CategoryOfReifications : Category SMME Reification.
-  admit.
-  Qed.
\ No newline at end of file
+(*
+ * 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 (enr_c_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 (enr_c_i s1) -> Reification s1 s2 (enr_c_i s2) -> Reification s0 s2 (enr_c_i s2).
+  intros.
+  refine
+    {| reification_rstar_f := reification_rstar_f X >>>> reification_rstar_f X0
+     ; reification_rstar   := PreMonoidalFunctorsCompose (reification_rstar X) (reification_rstar X0)
+     ; reification_r       := fun K => (reification_r X K) >>>> (reification_r X0 (enr_c_i s1))
+    |}.
+  intro K.
+  set (ni_associativity (reification_r X K) (reification_r X0 (enr_c_i s1)) (HomFunctor s2 (enr_c_i s2))) as q.
+  eapply ni_comp.
+  apply q.
+  clear q.
+  set (reification_commutes X K) as comm1.
+  set (reification_commutes X0 (enr_c_i s1)) as comm2.
+  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 (enr_c_i s1)) as c in *.
+  set (HomFunctor s2 (enr_c_i s2)) as c' in *.
+  set (HomFunctor s1 (enr_c_i s1)) as b' in *.
+  apply (ni_comp(F2:=b >>>> (b' >>>> x))).
+  apply (ni_respects1 b (c >>>> c') (b' >>>> x)).
+  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_respects2 (a >>>> a') (b >>>> b') x).
+  apply ni_inv.
+  apply comm1.
+  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.