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 MonoidalCategories_ch7_8.
+Require Import Coherence_ch7_8.
+Require Import Enrichment_ch2_8.
+Require Import RepresentableStructure_ch7_2.
+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
+ *
+ * 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.
+
+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_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.
+ 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.
projects / coq-hetmet.git / blobdiff