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.
Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
intros.
refine
- {| reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
+ {| 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.