1 (*********************************************************************************************************************************)
2 (* ReificationCategory: *)
4 (* There is a category whose objects are surjective monic monoidal enrichments (SMME's) and whose morphisms *)
7 (*********************************************************************************************************************************)
9 Generalizable All Variables.
10 Require Import Preamble.
11 Require Import General.
12 Require Import Categories_ch1_3.
13 Require Import Functors_ch1_4.
14 Require Import Isomorphisms_ch1_5.
15 Require Import ProductCategories_ch1_6_1.
16 Require Import OppositeCategories_ch1_6_2.
17 Require Import Enrichment_ch2_8.
18 Require Import Subcategories_ch7_1.
19 Require Import NaturalTransformations_ch7_4.
20 Require Import NaturalIsomorphisms_ch7_5.
21 Require Import MonoidalCategories_ch7_8.
22 Require Import Coherence_ch7_8.
23 Require Import Enrichment_ch2_8.
24 Require Import RepresentableStructure_ch7_2.
25 Require Import Reification.
26 Require Import WeakFunctorCategory.
27 Require Import SmallSMMEs.
29 Inductive ReificationOrIdentity : SMMEs -> SMMEs -> Type :=
30 | roi_id : forall smme:SMMEs, ReificationOrIdentity smme smme
31 | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (mon_i s2) -> ReificationOrIdentity s1 s2.
33 Definition reificationOrIdentityFunc
34 : forall s1 s2, ReificationOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
39 exists (reification_rstar_obj r).
43 Definition compose_reifications (s0 s1 s2:SMMEs) :
44 Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
47 {| reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
48 ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
51 set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
55 set (reification_commutes X K) as comm1.
56 set (reification_commutes X0 (mon_i s1)) as comm2.
60 Definition reificationOrIdentityComp
61 : forall s1 s2 s3, ReificationOrIdentity s1 s2 -> ReificationOrIdentity s2 s3 -> ReificationOrIdentity s1 s3.
66 apply (roi_reif _ _ r).
67 apply (roi_reif _ _ (compose_reifications _ _ _ r r0)).
70 Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
71 refine {| small_func := ReificationOrIdentity
72 ; small_func_id := fun s => roi_id s
73 ; small_func_func := fun smme1 smme2 f => projT2 (reificationOrIdentityFunc _ _ f)
74 ; small_func_comp := reificationOrIdentityComp
77 destruct f as [|fobj f]; simpl in *.
79 apply if_left_identity.
80 destruct g as [|gobj g]; simpl in *.
82 apply if_right_identity.
86 Definition CategoryOfReifications :=
87 WeakFunctorCategory MorphismsOfCategoryOfReifications.