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.
-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.
+ | 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, ReificationOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
+ : forall s1 s2 (f:ReificationOrIdentity s1 s2), Functor (enr_v s1) (enr_v s2) (reificationOrIdentityFobj s1 s2 f).
intros.
- destruct X.
- exists (fun x => x).
+ destruct f.
apply functor_id.
- exists (reification_rstar_obj r).
- apply r.
+ 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).
+ 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 := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
- ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
+ {| 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 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
+ 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 (mon_i s1)) as comm2.
- admit.
+ 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.
+ apply (reification_host_lang_pure _ _ _ X0).
Defined.
Definition reificationOrIdentityComp
Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
refine {| small_func := ReificationOrIdentity
; small_func_id := fun s => roi_id s
- ; small_func_func := fun smme1 smme2 f => projT2 (reificationOrIdentityFunc _ _ f)
+ ; small_func_func := fun smme1 smme2 f => reificationOrIdentityFunc _ _ f
; small_func_comp := reificationOrIdentityComp
|}; intros; simpl.
apply if_id.