Require Import Subcategories_ch7_1.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
+Require Import PreMonoidalCenter.
Require Import MonoidalCategories_ch7_8.
Require Import Coherence_ch7_8.
Require Import Enrichment_ch2_8.
Require Import GeneralizedArrow.
-Section ExtendFunctor.
-
- Context `(F:Functor).
- Context (P:c1 -> Prop).
-
- Definition domain_subcat := FullSubcategory c1 P.
-
- Definition functor_restricts_to_full_subcat_on_domain_fobj (a:domain_subcat) : c2 :=
- F (projT1 a).
-
- Definition functor_restricts_to_full_subcat_on_domain_fmor (a b:domain_subcat)(f:a~~{domain_subcat}~~>b) :
- (functor_restricts_to_full_subcat_on_domain_fobj a)~~{c2}~~>(functor_restricts_to_full_subcat_on_domain_fobj b) :=
- F \ (projT1 f).
-
- Lemma functor_restricts_to_full_subcat_on_domain : Functor domain_subcat c2 functor_restricts_to_full_subcat_on_domain_fobj.
- refine {| fmor := functor_restricts_to_full_subcat_on_domain_fmor |};
- unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
- setoid_rewrite H; reflexivity.
- setoid_rewrite fmor_preserves_id; reflexivity.
- setoid_rewrite <- fmor_preserves_comp; reflexivity.
- Defined.
-
-End ExtendFunctor.
-
-Section MonoidalSubCat.
-
- (* a monoidal subcategory is a full subcategory, closed under tensor and containing the unit object *)
- Class MonoidalSubCat {Ob}{Hom}{C:Category Ob Hom}{MFobj}{MF}{MI}(MC:MonoidalCat C MFobj MF MI) :=
- { msc_P : MC -> Prop
- ; msc_closed_under_tensor : forall o1 o2, msc_P o1 -> msc_P o2 -> msc_P (MC (pair_obj o1 o2))
- ; msc_contains_unit : msc_P (mon_i MC)
- ; msc_subcat := FullSubcategory MC msc_P
- }.
- Local Coercion msc_subcat : MonoidalSubCat >-> SubCategory.
-
- Context `(MSC:MonoidalSubCat).
-
- (* any full subcategory of a monoidal category, , is itself monoidal *)
- Definition mf_restricts_to_full_subcat_on_domain_fobj (a:MSC ×× MSC) : MSC.
- destruct a.
- destruct o.
- destruct o0.
- set (MC (pair_obj x x0)) as m'.
- exists m'.
- apply msc_closed_under_tensor; auto.
- Defined.
-
- Definition mf_restricts_to_full_subcat_on_domain_fmor
- {a}{b}
- (f:a~~{MSC ×× MSC}~~>b)
- :
- (mf_restricts_to_full_subcat_on_domain_fobj a)~~{MSC}~~>(mf_restricts_to_full_subcat_on_domain_fobj b).
- destruct a as [[a1 a1pf] [a2 a2pf]].
- destruct b as [[b1 b1pf] [b2 b2pf]].
- destruct f as [[f1 f1pf] [f2 f2pf]].
- simpl in *.
- exists (MC \ (pair_mor (pair_obj a1 a2) (pair_obj b1 b2) f1 f2)); auto.
- Defined.
-
- Lemma mf_restricts_to_full_subcat_on_domain : Functor (MSC ×× MSC) MSC
- mf_restricts_to_full_subcat_on_domain_fobj.
- refine {| fmor := fun a b f => mf_restricts_to_full_subcat_on_domain_fmor f |};
- unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
- admit.
- admit.
- admit.
- Defined.
-
- Definition subcat_i : MSC.
- exists (mon_i MC).
- apply msc_contains_unit.
- Defined.
-
- Lemma full_subcat_is_monoidal : MonoidalCat MSC _ mf_restricts_to_full_subcat_on_domain subcat_i.
- admit.
- Defined.
-
- Lemma inclusion_functor_monoidal : MonoidalFunctor full_subcat_is_monoidal MC (InclusionFunctor _ MSC).
- admit.
- Defined.
-
-End MonoidalSubCat.
-Coercion full_subcat_is_monoidal : MonoidalSubCat >-> MonoidalCat.
-
Section ArrowInLanguage.
(* an Arrow In A Programming Language consists of... *)
(* a host language: *)
- Context (Host:ProgrammingLanguageSMME).
+ Context `(Host:ProgrammingLanguage).
(* ... for which Types(L) is cartesian: *)
- Context {MF}(center_of_TypesL:MonoidalCat (TypesL _ _ Host) (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []).
+ Context (center_of_TypesL:MonoidalCat (TypesL_PreMonoidal Host)).
- (* along with a monoidal subcategory of Z(Types(L)) *)
- Context (VK:MonoidalSubCat center_of_TypesL).
-
- (* a premonoidal category enriched in aforementioned full subcategory *)
- Context {Kehom:center_of_TypesL -> center_of_TypesL -> VK}.
+ (* along with a full subcategory of Z(Types(L)) *)
+ Context {P}(VK:FullSubcategory (Center (TypesL_PreMonoidal Host)) P).
- Context {KE:@ECategory _ _ VK _ _ _ VK center_of_TypesL Kehom}.
+ Context (Pobj_unit : P []).
+ Context (Pobj_closed : forall a b, P a → P b → P (bin_obj(BinoidalCat:=Center_is_PreMonoidal (TypesL_PreMonoidal Host)) a b)).
+ Definition VKM :=
+ PreMonoidalFullSubcategory_PreMonoidal (Center_is_PreMonoidal (TypesL_PreMonoidal Host)) VK Pobj_unit Pobj_closed.
- Context (CC:CartesianCat center_of_TypesL).
-
- Context {kbo}{kbc}(K:@PreMonoidalCat _ _ KE kbo kbc (@one _ _ _ (car_terminal(CartesianCat:=CC)))).
+ (* a premonoidal category enriched in aforementioned full subcategory *)
+ Context (Kehom:(Center (TypesL_PreMonoidal Host)) -> (Center (TypesL_PreMonoidal Host)) -> @ob _ _ VK).
+ Context (KE :@ECategory (@ob _ _ VK) (@hom _ _ VK) VK _ VKM (pmon_I VKM) VKM (Center (TypesL_PreMonoidal Host)) Kehom).
+ Context {kbo :(Center (TypesL_PreMonoidal Host)) -> (Center (TypesL_PreMonoidal Host)) -> (Center (TypesL_PreMonoidal Host))}.
+ Context (kbc :@BinoidalCat (Center (TypesL_PreMonoidal Host)) _ (Underlying KE) kbo).
- (*
- Definition K_enrichment : Enrichment.
- refine {| enr_c := KE |}.
- Defined.
- Context (K_surjective:SurjectiveEnrichment K_enrichment).
- *)
+ Definition one' := @one _ _ _ (car_terminal(CartesianCat:=CC))
+ Context (K :@PreMonoidalCat _ _ KE kbo kbc ).
+ Context (CC:CartesianCat (Center (TypesL_PreMonoidal Host))).
Definition ArrowInProgrammingLanguage :=
- @FreydCategory _ _ _ _ _ _ center_of_TypesL _ _ _ _ K.
+ @FreydCategory _ _ _ _ _ _ (Center (TypesL_PreMonoidal Host)) _ _ _ _ K.
Definition ArrowsAreGeneralizedArrows (arrow:ArrowInProgrammingLanguage) : GeneralizedArrow K_enrichment Host.
refine {| ga_functor_monoidal := inclusion_functor_monoidal VK |}.
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