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February 16, 2017 23:24
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universe variables u v | |
structure Category := | |
(Obj : Type u) | |
(Hom : Obj → Obj → Type v) | |
universe variables u1 v1 u2 v2 | |
structure Functor (C : Category.{ u1 v1 }) (D : Category.{ u2 v2 }) := | |
(onObjects : C^.Obj → D^.Obj) | |
@[reducible] definition ProductCategory (C : Category) (D : Category) : | |
Category := | |
{ | |
Obj := C^.Obj × D^.Obj, | |
Hom := (λ X Y : C^.Obj × D^.Obj, C^.Hom (X^.fst) (Y^.fst) × D^.Hom (X^.snd) (Y^.snd)) | |
} | |
namespace ProductCategory | |
notation C `×` D := ProductCategory C D | |
end ProductCategory | |
@[reducible] definition TensorProduct ( C: Category ) := Functor ( C × C ) C | |
structure MonoidalCategory | |
extends carrier : Category := | |
(tensor : TensorProduct carrier) | |
instance MonoidalCategory_coercion : has_coe MonoidalCategory Category := | |
⟨MonoidalCategory.to_Category⟩ | |
-- Convenience methods which take two arguments, rather than a pair. (This seems to often help the elaborator avoid getting stuck on `prod.mk`.) | |
@[reducible] definition MonoidalCategory.tensorObjects { C : MonoidalCategory } ( X Y : C^.Obj ) : C^.Obj := C^.tensor^.onObjects (X, Y) | |
definition tensor_on_left { C: MonoidalCategory.{u v} } ( Z: C^.Obj ) : Functor.{u v u v} C C := | |
{ | |
onObjects := λ X, C^.tensorObjects Z X | |
} |
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