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 reificationOrIdentityFobj s1 s2 (f:ReificationOrIdentity s1 s2) : s1 -> s2 :=
35 | roi_id s => (fun x => x)
36 | roi_reif s1 s2 f => reification_rstar_obj f
39 Definition reificationOrIdentityFunc
40 : forall s1 s2 (f:ReificationOrIdentity s1 s2), Functor (enr_v s1) (enr_v s2) (reificationOrIdentityFobj s1 s2 f).
44 unfold reificationOrIdentityFobj.
45 apply (reification_rstar_f r).
48 Definition compose_reifications (s0 s1 s2:SMMEs) :
49 Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
52 {| reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
53 ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
56 set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
60 set (reification_commutes X K) as comm1.
61 set (reification_commutes X0 (mon_i s1)) as comm2.
62 set (RepresentableFunctor s0 K) as a in *.
63 set (reification_rstar_f X) as a' in *.
64 set (reification_rstar_f X0) as x in *.
65 set (reification_r X K) as b in *.
66 set (reification_r X0 (mon_i s1)) as c in *.
67 set (RepresentableFunctor s2 (mon_i s2)) as c' in *.
68 set (RepresentableFunctor s1 (mon_i s1)) as b' in *.
69 apply (ni_comp(F2:=b >>>> (b' >>>> x))).
70 apply (@ni_respects _ _ _ _ _ _ _ _ _ _ b _ b _ (c >>>> c') _ (b' >>>> x)).
75 apply (ni_associativity b b' x).
79 apply (ni_associativity a a' x).
80 apply (@ni_respects _ _ _ _ _ _ _ _ _ _ (a >>>> a') _ (b >>>> b') _ x _ x).
86 Definition reificationOrIdentityComp
87 : forall s1 s2 s3, ReificationOrIdentity s1 s2 -> ReificationOrIdentity s2 s3 -> ReificationOrIdentity s1 s3.
92 apply (roi_reif _ _ r).
93 apply (roi_reif _ _ (compose_reifications _ _ _ r r0)).
96 Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
97 refine {| small_func := ReificationOrIdentity
98 ; small_func_id := fun s => roi_id s
99 ; small_func_func := fun smme1 smme2 f => reificationOrIdentityFunc _ _ f
100 ; small_func_comp := reificationOrIdentityComp
103 destruct f as [|fobj f]; simpl in *.
105 apply if_left_identity.
106 destruct g as [|gobj g]; simpl in *.
108 apply if_right_identity.
112 Definition CategoryOfReifications :=
113 WeakFunctorCategory MorphismsOfCategoryOfReifications.