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 (* Technically reifications form merely a *semicategory* (no identity
30 * maps), but one can always freely adjoin identity maps (and nothing
31 * else) to a semicategory to get a category whose non-identity-map
32 * portion is identical to the original semicategory
34 * Also, technically this category has ALL enrichments (not just the
35 * surjective monic monoidal ones), though there maps OUT OF only the
36 * surjective enrichments and INTO only the monic monoidal
37 * enrichments. It's a big pain to do this in Coq, but sort of might
38 * matter in real life: a language with really severe substructural
39 * restrictions might fail to be monoidally enriched, meaning we can't
40 * use it as a host language. But that's for the next paper...
42 Inductive ReificationOrIdentity : SMMEs -> SMMEs -> Type :=
43 | roi_id : forall smme:SMMEs, ReificationOrIdentity smme smme
44 | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (mon_i s2) -> ReificationOrIdentity s1 s2.
46 Definition reificationOrIdentityFobj s1 s2 (f:ReificationOrIdentity s1 s2) : s1 -> s2 :=
48 | roi_id s => (fun x => x)
49 | roi_reif s1 s2 f => reification_rstar_obj f
52 Definition reificationOrIdentityFunc
53 : forall s1 s2 (f:ReificationOrIdentity s1 s2), Functor (enr_v s1) (enr_v s2) (reificationOrIdentityFobj s1 s2 f).
57 unfold reificationOrIdentityFobj.
58 apply (reification_rstar_f r).
61 Definition compose_reifications (s0 s1 s2:SMMEs) :
62 Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
65 {| reification_rstar_f := reification_rstar_f X >>>> reification_rstar_f X0
66 ; reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
67 ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
70 set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
74 set (reification_commutes X K) as comm1.
75 set (reification_commutes X0 (mon_i s1)) as comm2.
76 set (RepresentableFunctor s0 K) as a in *.
77 set (reification_rstar_f X) as a' in *.
78 set (reification_rstar_f X0) as x in *.
79 set (reification_r X K) as b in *.
80 set (reification_r X0 (mon_i s1)) as c in *.
81 set (RepresentableFunctor s2 (mon_i s2)) as c' in *.
82 set (RepresentableFunctor s1 (mon_i s1)) as b' in *.
83 apply (ni_comp(F2:=b >>>> (b' >>>> x))).
84 apply (@ni_respects _ _ _ _ _ _ _ _ _ _ b _ b _ (c >>>> c') _ (b' >>>> x)).
89 apply (ni_associativity b b' x).
93 apply (ni_associativity a a' x).
94 apply (@ni_respects _ _ _ _ _ _ _ _ _ _ (a >>>> a') _ (b >>>> b') _ x _ x).
100 Definition reificationOrIdentityComp
101 : forall s1 s2 s3, ReificationOrIdentity s1 s2 -> ReificationOrIdentity s2 s3 -> ReificationOrIdentity s1 s3.
106 apply (roi_reif _ _ r).
107 apply (roi_reif _ _ (compose_reifications _ _ _ r r0)).
110 Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
111 refine {| small_func := ReificationOrIdentity
112 ; small_func_id := fun s => roi_id s
113 ; small_func_func := fun smme1 smme2 f => reificationOrIdentityFunc _ _ f
114 ; small_func_comp := reificationOrIdentityComp
117 destruct f as [|fobj f]; simpl in *.
119 apply if_left_identity.
120 destruct g as [|gobj g]; simpl in *.
122 apply if_right_identity.
126 Definition CategoryOfReifications :=
127 WeakFunctorCategory MorphismsOfCategoryOfReifications.