1 (*********************************************************************************************************************************)
2 (* CategoryOfGeneralizedArrows: *)
4 (* There is a category whose objects are surjective monic monoidal enrichments (SMME's) and whose morphisms *)
5 (* are generalized Arrows *)
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 GeneralizedArrow.
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
35 * Also, technically this category has ALL enrichments (not just the
36 * surjective monic monoidal ones), though there maps OUT OF only the
37 * surjective enrichments and INTO only the monic monoidal
38 * enrichments. It's a big pain to do this in Coq, but sort of might
39 * matter in real life: a language with really severe substructural
40 * restrictions might fail to be monoidally enriched, meaning we can't
41 * use it as a host language. But that's for the next paper...
43 Inductive GeneralizedArrowOrIdentity : SMMEs -> SMMEs -> Type :=
44 | gaoi_id : forall smme:SMMEs, GeneralizedArrowOrIdentity smme smme
45 | gaoi_ga : forall s1 s2:SMMEs, GeneralizedArrow s1 s2 -> GeneralizedArrowOrIdentity s1 s2.
47 Definition generalizedArrowOrIdentityFobj (s1 s2:SMMEs) (f:GeneralizedArrowOrIdentity s1 s2) : s1 -> s2 :=
48 match f in GeneralizedArrowOrIdentity S1 S2 return S1 -> S2 with
49 | gaoi_id s => fun x => x
50 | gaoi_ga s1 s2 f => fun a => ehom(ECategory:=s2) (mon_i (smme_mon s2)) (ga_functor_obj f a)
53 Definition generalizedArrowOrIdentityFunc s1 s2 (f:GeneralizedArrowOrIdentity s1 s2)
54 : Functor s1 s2 (generalizedArrowOrIdentityFobj _ _ f) :=
56 | gaoi_id s => functor_id _
57 | gaoi_ga s1 s2 f => ga_functor f >>>> RepresentableFunctor s2 (mon_i s2)
60 Definition compose_generalizedArrows (s0 s1 s2:SMMEs) :
61 GeneralizedArrow s0 s1 -> GeneralizedArrow s1 s2 -> GeneralizedArrow s0 s2.
65 {| ga_functor := g01 >>>> RepresentableFunctor s1 (mon_i s1) >>>> g12 |}.
66 apply MonoidalFunctorsCompose.
67 apply MonoidalFunctorsCompose.
68 apply (ga_functor_monoidal g01).
70 apply (ga_functor_monoidal g12).
73 Definition generalizedArrowOrIdentityComp
74 : forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3.
79 apply (gaoi_ga _ _ g).
80 apply (gaoi_ga _ _ (compose_generalizedArrows _ _ _ g g0)).
83 Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs.
84 refine {| small_func := GeneralizedArrowOrIdentity
85 ; small_func_id := fun s => gaoi_id s
86 ; small_func_func := fun smme1 smme2 f => generalizedArrowOrIdentityFunc _ _ f
87 ; small_func_comp := generalizedArrowOrIdentityComp
90 destruct f as [|fobj f]; simpl in *.
92 apply if_left_identity.
93 destruct g as [|gobj g]; simpl in *.
95 apply if_right_identity.
99 apply if_associativity.
102 Definition CategoryOfGeneralizedArrows :=
103 WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.