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.
-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 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 generalizedArrowOrIdentityFunc
- : forall s1 s2, GeneralizedArrowOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
- intros.
- destruct X.
- exists (fun x => x).
- apply functor_id.
- eapply existT.
- apply (g >>>> RepresentableFunctor s2 (mon_i s2)).
- Defined.
+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 compose_generalizedArrows (s0 s1 s2:SMMEs) :
- GeneralizedArrow s0 s1 -> GeneralizedArrow s1 s2 -> GeneralizedArrow s0 s2.
- intro g01.
- intro g12.
- refine
- {| ga_functor := g01 >>>> RepresentableFunctor s1 (mon_i s1) >>>> g12 |}.
- apply MonoidalFunctorsCompose.
- apply MonoidalFunctorsCompose.
- apply (ga_functor_monoidal g01).
- apply (me_mf s1).
- apply (ga_functor_monoidal g12).
- Defined.
+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 }.
+ apply ga_host_lang_pure.
+ Defined.
Definition generalizedArrowOrIdentityComp
: forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3.
Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs.
refine {| small_func := GeneralizedArrowOrIdentity
; small_func_id := fun s => gaoi_id s
- ; small_func_func := fun smme1 smme2 f => projT2 (generalizedArrowOrIdentityFunc _ _ f)
+ ; small_func_func := fun smme1 smme2 f => generalizedArrowOrIdentityFunc _ _ f
; small_func_comp := generalizedArrowOrIdentityComp
|}; intros; simpl.
apply if_id.
apply if_right_identity.
unfold mf_F.
idtac.
- unfold mf_f.
apply if_associativity.
Defined.