(*********************************************************************************************************************************) (* ReificationCategory: *) (* *) (* There is a category whose objects are surjective monic monoidal enrichments (SMME's) and whose morphisms *) (* are reifications *) (* *) (*********************************************************************************************************************************) 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 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. (* * 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 (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 (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 (enr_c_i s1) -> Reification s1 s2 (enr_c_i s2) -> Reification s0 s2 (enr_c_i s2). intros. refine {| 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 (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 (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 : 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.