(*********************************************************************************************************************************) (* CategoryOfGeneralizedArrows: *) (* *) (* There is a category whose objects are surjective monic monoidal enrichments (SMME's) and whose morphisms *) (* are generalized Arrows *) (* *) (*********************************************************************************************************************************) Generalizable All Variables. Require Import Preamble. Require Import General. 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 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 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) (mon_i (smme_mon s2)) (ga_functor_obj f a) end. 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 (mon_i s2) 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 >>>> HomFunctor 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 generalizedArrowOrIdentityComp : forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3. intros. destruct X. apply X0. destruct X0. apply (gaoi_ga _ _ g). apply (gaoi_ga _ _ (compose_generalizedArrows _ _ _ g g0)). Defined. Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs. refine {| small_func := GeneralizedArrowOrIdentity ; small_func_id := fun s => gaoi_id s ; small_func_func := fun smme1 smme2 f => generalizedArrowOrIdentityFunc _ _ f ; small_func_comp := generalizedArrowOrIdentityComp |}; 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. unfold mf_F. idtac. unfold mf_f. apply if_associativity. Defined. Definition CategoryOfGeneralizedArrows := WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.