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.
29 Inductive GeneralizedArrowOrIdentity : SMMEs -> SMMEs -> Type :=
30 | gaoi_id : forall smme:SMMEs, GeneralizedArrowOrIdentity smme smme
31 | gaoi_ga : forall s1 s2:SMMEs, GeneralizedArrow s1 s2 -> GeneralizedArrowOrIdentity s1 s2.
33 Definition generalizedArrowOrIdentityFunc
34 : forall s1 s2, GeneralizedArrowOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
40 apply (g >>>> RepresentableFunctor s2 (mon_i s2)).
43 Definition compose_generalizedArrows (s0 s1 s2:SMMEs) :
44 GeneralizedArrow s0 s1 -> GeneralizedArrow s1 s2 -> GeneralizedArrow s0 s2.
48 {| ga_functor := g01 >>>> RepresentableFunctor s1 (mon_i s1) >>>> g12 |}.
49 apply MonoidalFunctorsCompose.
50 apply MonoidalFunctorsCompose.
51 apply (ga_functor_monoidal g01).
53 apply (ga_functor_monoidal g12).
56 Definition generalizedArrowOrIdentityComp
57 : forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3.
62 apply (gaoi_ga _ _ g).
63 apply (gaoi_ga _ _ (compose_generalizedArrows _ _ _ g g0)).
66 Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs.
67 refine {| small_func := GeneralizedArrowOrIdentity
68 ; small_func_id := fun s => gaoi_id s
69 ; small_func_func := fun smme1 smme2 f => projT2 (generalizedArrowOrIdentityFunc _ _ f)
70 ; small_func_comp := generalizedArrowOrIdentityComp
73 destruct f as [|fobj f]; simpl in *.
75 apply if_left_identity.
76 destruct g as [|gobj g]; simpl in *.
78 apply if_right_identity.
82 apply if_associativity.
85 Definition CategoryOfGeneralizedArrows :=
86 WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.