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 NaturalDeduction.
Require Import NaturalDeductionCategory.
-Require Import ProgrammingLanguage.
-Require Import ProgrammingLanguageGeneralizedArrow.
+Require Import ProgrammingLanguageCategory.
Require Import FreydCategories.
-
+Require Import Enrichments.
Require Import GeneralizedArrow.
Section ArrowInLanguage.
(* an Arrow In A Programming Language consists of... *)
(* a host language: *)
- 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 `(Host : ProgrammingLanguage).
- (* along with a monoidal subcategory of Z(Types(L)) *)
- Context (VK:MonoidalSubCat center_of_TypesL).
+ Context (Host_Monoidal : MonoidalCat (TypesL_PreMonoidal Host)).
+ Context (Host_Cartesian : CartesianCat Host_Monoidal).
- (* a premonoidal category enriched in aforementioned full subcategory *)
- Context (Kehom:center_of_TypesL -> center_of_TypesL -> @ob _ _ VK).
+ Context
+ {P}
+ (Pobj_unit : P [])
+ (Pobj_closed : forall a b, P a → P b → P (bin_obj(BinoidalCat:=Center_is_PreMonoidal (TypesL_PreMonoidal Host)) a b)).
- Context (KE:@ECategory (@ob _ _ VK) (@hom _ _ VK) VK _ VK
- (mon_i (full_subcat_is_monoidal VK)) (full_subcat_is_monoidal VK) center_of_TypesL Kehom).
+ Context (VK : FullSubcategory Host_Cartesian P).
- Context {kbo:center_of_TypesL -> center_of_TypesL -> center_of_TypesL}.
+ Context ehom KE (K_bin:@EBinoidalCat _ _ VK _ _ _
+ (PreMonoidalFullSubcategory_PreMonoidal Host_Cartesian VK Pobj_unit Pobj_closed)
+ (TypesL Host) ehom KE (bin_obj(BinoidalCat:=Host_Monoidal))).
- Context (kbc:@BinoidalCat center_of_TypesL _ (Underlying KE) kbo).
+ Context (K_premonoidal:PreMonoidalCat K_bin (one(TerminalObject:=Host_Cartesian))).
- Check (@PreMonoidalCat)
- Definition one' := @one _ _ _ (car_terminal(CartesianCat:=CC))
- Context (K:@PreMonoidalCat _ _ KE kbo kbc ).
- Context (CC:CartesianCat center_of_TypesL).
+ Definition ArrowInProgrammingLanguage :=
+ @FreydCategory _ _ _ _ _ _ _ _ Host_Cartesian _ _ K_bin K_premonoidal.
- (*
Definition K_enrichment : Enrichment.
- refine {| enr_c := KE |}.
- Defined.
- Context (K_surjective:SurjectiveEnrichment K_enrichment).
- *)
-
- Definition ArrowInProgrammingLanguage :=
- @FreydCategory _ _ _ _ _ _ center_of_TypesL _ _ _ _ K.
+ refine
+ {| enr_c_pm := K_premonoidal
+ ; enr_v_mon := MonoidalFullSubcategory_Monoidal Host_Cartesian _ _ VK
+ |}.
+ Defined.
- Definition ArrowsAreGeneralizedArrows (arrow:ArrowInProgrammingLanguage) : GeneralizedArrow K_enrichment Host.
- refine {| ga_functor_monoidal := inclusion_functor_monoidal VK |}.
- Defined.
+ Instance ArrowsAreGeneralizedArrows : GeneralizedArrow K_enrichment (TypesEnrichedInJudgments Host) :=
+ { ga_functor_monoidal :=
+ PreMonoidalFullSubcategoryInclusionFunctor_PreMonoidal Host_Cartesian VK Pobj_unit Pobj_closed Host_Cartesian }.
-End GArrowInLanguage.
+End ArrowInLanguage.