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.
30 * Technically reifications form merely a *semicategory* (no identity
31 * maps), but one can always freely adjoin identity maps (and nothing
32 * else) to a semicategory to get a category whose non-identity-map
33 * portion is identical to the original semicategory (closing under
34 * composition after putting in the identity maps never produces any
37 * Also, technically this category has ALL enrichments (not just the
38 * surjective monic monoidal ones), though there maps OUT OF only the
39 * surjective enrichments and INTO only the monic monoidal
40 * enrichments. It's a big pain to do this in Coq, but sort of might
41 * matter in real life: a language with really severe substructural
42 * restrictions might fail to be monoidally enriched, meaning we can't
43 * use it as a host language. But that's for the next paper...
45 Inductive ReificationOrIdentity : SMMEs -> SMMEs -> Type :=
46 | roi_id : forall smme:SMMEs, ReificationOrIdentity smme smme
47 | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (mon_i s2) -> ReificationOrIdentity s1 s2.
49 Definition reificationOrIdentityFobj s1 s2 (f:ReificationOrIdentity s1 s2) : s1 -> s2 :=
51 | roi_id s => (fun x => x)
52 | roi_reif s1 s2 f => reification_rstar_obj f
55 Definition reificationOrIdentityFunc
56 : forall s1 s2 (f:ReificationOrIdentity s1 s2), Functor (enr_v s1) (enr_v s2) (reificationOrIdentityFobj s1 s2 f).
60 unfold reificationOrIdentityFobj.
61 apply (reification_rstar_f r).
64 Definition compose_reifications (s0 s1 s2:SMMEs) :
65 Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
68 {| reification_rstar_f := reification_rstar_f X >>>> reification_rstar_f X0
69 ; reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
70 ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
73 set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (HomFunctor s2 (mon_i s2))) as q.
77 set (reification_commutes X K) as comm1.
78 set (reification_commutes X0 (mon_i s1)) as comm2.
79 set (HomFunctor s0 K) as a in *.
80 set (reification_rstar_f X) as a' in *.
81 set (reification_rstar_f X0) as x in *.
82 set (reification_r X K) as b in *.
83 set (reification_r X0 (mon_i s1)) as c in *.
84 set (HomFunctor s2 (mon_i s2)) as c' in *.
85 set (HomFunctor s1 (mon_i s1)) as b' in *.
86 apply (ni_comp(F2:=b >>>> (b' >>>> x))).
87 apply (@ni_respects _ _ _ _ _ _ _ _ _ _ b _ b _ (c >>>> c') _ (b' >>>> x)).
92 apply (ni_associativity b b' x).
96 apply (ni_associativity a a' x).
97 apply (@ni_respects _ _ _ _ _ _ _ _ _ _ (a >>>> a') _ (b >>>> b') _ x _ x).
103 Definition reificationOrIdentityComp
104 : forall s1 s2 s3, ReificationOrIdentity s1 s2 -> ReificationOrIdentity s2 s3 -> ReificationOrIdentity s1 s3.
109 apply (roi_reif _ _ r).
110 apply (roi_reif _ _ (compose_reifications _ _ _ r r0)).
113 Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
114 refine {| small_func := ReificationOrIdentity
115 ; small_func_id := fun s => roi_id s
116 ; small_func_func := fun smme1 smme2 f => reificationOrIdentityFunc _ _ f
117 ; small_func_comp := reificationOrIdentityComp
120 destruct f as [|fobj f]; simpl in *.
122 apply if_left_identity.
123 destruct g as [|gobj g]; simpl in *.
125 apply if_right_identity.
129 Definition CategoryOfReifications :=
130 WeakFunctorCategory MorphismsOfCategoryOfReifications.