Closed monoidal categories #
Define (right) closed objects and (right) closed monoidal categories.
TODO #
Some of the theorems proved about cartesian closed categories should be generalised and moved to this file.
class
CategoryTheory.Closed
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(X : C)
:
Type (max u v)
a choice of a left adjoint for
tensorLeft X
An object X is (right) closed if (X ⊗ -) is a left adjoint.
Instances
class
CategoryTheory.MonoidalClosed
(C : Type u)
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
:
Type (max u v)
- closed : (X : C) → CategoryTheory.Closed X
A monoidal category C is (right) monoidal closed if every object is (right) closed.
Instances
def
CategoryTheory.tensorClosed
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{X : C}
{Y : C}
(hX : CategoryTheory.Closed X)
(hY : CategoryTheory.Closed Y)
:
If X and Y are closed then X ⊗ Y is.
This isn't an instance because it's not usually how we want to construct internal homs,
we'll usually prove all objects are closed uniformly.
Instances For
def
CategoryTheory.unitClosed
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
:
The unit object is always closed. This isn't an instance because most of the time we'll prove closedness for all objects at once, rather than just for this one.
Instances For
def
CategoryTheory.ihom
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
:
This is the internal hom A ⟶[C] -.
Instances For
def
CategoryTheory.ihom.adjunction
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
:
The adjunction between A ⊗ - and A ⟹ -.
Instances For
def
CategoryTheory.ihom.ev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
:
The evaluation natural transformation.
Instances For
def
CategoryTheory.ihom.coev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
:
The coevaluation natural transformation.
Instances For
@[simp]
theorem
CategoryTheory.ihom.ihom_adjunction_counit
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
:
(CategoryTheory.ihom.adjunction A).counit = CategoryTheory.ihom.ev A
@[simp]
theorem
CategoryTheory.ihom.ihom_adjunction_unit
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
:
@[simp]
theorem
CategoryTheory.ihom.ev_naturality_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
{X : C}
{Y : C}
(f : X ⟶ Y)
{Z : C}
(h : (CategoryTheory.Functor.id C).obj Y ⟶ Z)
:
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id A) ((CategoryTheory.ihom A).map f))
(CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.ev A).app Y) h) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.ev A).app X) (CategoryTheory.CategoryStruct.comp f h)
@[simp]
theorem
CategoryTheory.ihom.ev_naturality
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
{X : C}
{Y : C}
(f : X ⟶ Y)
:
@[simp]
theorem
CategoryTheory.ihom.coev_naturality_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
{X : C}
{Y : C}
(f : X ⟶ Y)
{Z : C}
(h : (CategoryTheory.Functor.comp (CategoryTheory.MonoidalCategory.tensorLeft A) (CategoryTheory.ihom A)).obj Y ⟶ Z)
:
CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app Y) h) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app X)
(CategoryTheory.CategoryStruct.comp
((CategoryTheory.ihom A).map (CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id A) f))
h)
@[simp]
theorem
CategoryTheory.ihom.coev_naturality
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
[CategoryTheory.Closed A]
{X : C}
{Y : C}
(f : X ⟶ Y)
:
A ⟶[C] B denotes the internal hom from A to B
Instances For
@[simp]
theorem
CategoryTheory.ihom.ev_coev_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
(B : C)
[CategoryTheory.Closed A]
{Z : C}
(h : (CategoryTheory.Functor.id C).obj (CategoryTheory.MonoidalCategory.tensorObj A B) ⟶ Z)
:
@[simp]
theorem
CategoryTheory.ihom.ev_coev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
(B : C)
[CategoryTheory.Closed A]
:
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id A)
((CategoryTheory.ihom.coev A).app B))
((CategoryTheory.ihom.ev A).app (CategoryTheory.MonoidalCategory.tensorObj A B)) = CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj A B)
@[simp]
theorem
CategoryTheory.ihom.coev_ev_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
(B : C)
[CategoryTheory.Closed A]
{Z : C}
(h : (CategoryTheory.ihom A).obj ((CategoryTheory.Functor.id C).obj B) ⟶ Z)
:
CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app ((CategoryTheory.ihom A).obj B))
(CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom A).map ((CategoryTheory.ihom.ev A).app B)) h) = h
@[simp]
theorem
CategoryTheory.ihom.coev_ev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
(B : C)
[CategoryTheory.Closed A]
:
CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app ((CategoryTheory.ihom A).obj B))
((CategoryTheory.ihom A).map ((CategoryTheory.ihom.ev A).app B)) = CategoryTheory.CategoryStruct.id ((CategoryTheory.ihom A).obj B)
def
CategoryTheory.MonoidalClosed.curry
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
:
(CategoryTheory.MonoidalCategory.tensorObj A Y ⟶ X) → (Y ⟶ (CategoryTheory.ihom A).obj X)
Currying in a monoidal closed category.
Instances For
def
CategoryTheory.MonoidalClosed.uncurry
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
:
(Y ⟶ (CategoryTheory.ihom A).obj X) → (CategoryTheory.MonoidalCategory.tensorObj A Y ⟶ X)
Uncurrying in a monoidal closed category.
Instances For
@[simp]
theorem
CategoryTheory.MonoidalClosed.homEquiv_apply_eq
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(f : CategoryTheory.MonoidalCategory.tensorObj A Y ⟶ X)
:
@[simp]
theorem
CategoryTheory.MonoidalClosed.homEquiv_symm_apply_eq
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(f : Y ⟶ (CategoryTheory.ihom A).obj X)
:
theorem
CategoryTheory.MonoidalClosed.curry_natural_left_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{X' : C}
{Y : C}
[CategoryTheory.Closed A]
(f : X ⟶ X')
(g : CategoryTheory.MonoidalCategory.tensorObj A X' ⟶ Y)
{Z : C}
(h : (CategoryTheory.ihom A).obj Y ⟶ Z)
:
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalClosed.curry
(CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id A) f) g))
h = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalClosed.curry g) h)
theorem
CategoryTheory.MonoidalClosed.curry_natural_left
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{X' : C}
{Y : C}
[CategoryTheory.Closed A]
(f : X ⟶ X')
(g : CategoryTheory.MonoidalCategory.tensorObj A X' ⟶ Y)
:
theorem
CategoryTheory.MonoidalClosed.curry_natural_right_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
{Y' : C}
[CategoryTheory.Closed A]
(f : CategoryTheory.MonoidalCategory.tensorObj A X ⟶ Y)
(g : Y ⟶ Y')
{Z : C}
(h : (CategoryTheory.ihom A).obj Y' ⟶ Z)
:
theorem
CategoryTheory.MonoidalClosed.curry_natural_right
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
{Y' : C}
[CategoryTheory.Closed A]
(f : CategoryTheory.MonoidalCategory.tensorObj A X ⟶ Y)
(g : Y ⟶ Y')
:
theorem
CategoryTheory.MonoidalClosed.uncurry_natural_right_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
{Y' : C}
[CategoryTheory.Closed A]
(f : X ⟶ (CategoryTheory.ihom A).obj Y)
(g : Y ⟶ Y')
{Z : C}
(h : Y' ⟶ Z)
:
theorem
CategoryTheory.MonoidalClosed.uncurry_natural_right
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
{Y' : C}
[CategoryTheory.Closed A]
(f : X ⟶ (CategoryTheory.ihom A).obj Y)
(g : Y ⟶ Y')
:
theorem
CategoryTheory.MonoidalClosed.uncurry_natural_left_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{X' : C}
{Y : C}
[CategoryTheory.Closed A]
(f : X ⟶ X')
(g : X' ⟶ (CategoryTheory.ihom A).obj Y)
{Z : C}
(h : Y ⟶ Z)
:
CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalClosed.uncurry (CategoryTheory.CategoryStruct.comp f g)) h = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id A) f)
(CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalClosed.uncurry g) h)
theorem
CategoryTheory.MonoidalClosed.uncurry_natural_left
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{X' : C}
{Y : C}
[CategoryTheory.Closed A]
(f : X ⟶ X')
(g : X' ⟶ (CategoryTheory.ihom A).obj Y)
:
@[simp]
theorem
CategoryTheory.MonoidalClosed.uncurry_curry
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(f : CategoryTheory.MonoidalCategory.tensorObj A X ⟶ Y)
:
@[simp]
theorem
CategoryTheory.MonoidalClosed.curry_uncurry
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(f : X ⟶ (CategoryTheory.ihom A).obj Y)
:
theorem
CategoryTheory.MonoidalClosed.curry_eq_iff
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(f : CategoryTheory.MonoidalCategory.tensorObj A Y ⟶ X)
(g : Y ⟶ (CategoryTheory.ihom A).obj X)
:
theorem
CategoryTheory.MonoidalClosed.eq_curry_iff
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(f : CategoryTheory.MonoidalCategory.tensorObj A Y ⟶ X)
(g : Y ⟶ (CategoryTheory.ihom A).obj X)
:
theorem
CategoryTheory.MonoidalClosed.uncurry_eq
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(g : Y ⟶ (CategoryTheory.ihom A).obj X)
:
theorem
CategoryTheory.MonoidalClosed.curry_eq
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
(g : CategoryTheory.MonoidalCategory.tensorObj A Y ⟶ X)
:
CategoryTheory.MonoidalClosed.curry g = CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app Y) ((CategoryTheory.ihom A).map g)
theorem
CategoryTheory.MonoidalClosed.curry_injective
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
:
Function.Injective CategoryTheory.MonoidalClosed.curry
theorem
CategoryTheory.MonoidalClosed.uncurry_injective
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{X : C}
{Y : C}
[CategoryTheory.Closed A]
:
Function.Injective CategoryTheory.MonoidalClosed.uncurry
theorem
CategoryTheory.MonoidalClosed.uncurry_id_eq_ev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
(X : C)
[CategoryTheory.Closed A]
:
CategoryTheory.MonoidalClosed.uncurry (CategoryTheory.CategoryStruct.id ((CategoryTheory.ihom A).obj X)) = (CategoryTheory.ihom.ev A).app X
theorem
CategoryTheory.MonoidalClosed.curry_id_eq_coev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
(A : C)
(X : C)
[CategoryTheory.Closed A]
:
def
CategoryTheory.MonoidalClosed.pre
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{B : C}
[CategoryTheory.Closed A]
[CategoryTheory.Closed B]
(f : B ⟶ A)
:
Pre-compose an internal hom with an external hom.
Instances For
@[simp]
theorem
CategoryTheory.MonoidalClosed.id_tensor_pre_app_comp_ev_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{B : C}
[CategoryTheory.Closed A]
[CategoryTheory.Closed B]
(f : B ⟶ A)
(X : C)
{Z : C}
(h : (CategoryTheory.Functor.id C).obj X ⟶ Z)
:
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id B)
((CategoryTheory.MonoidalClosed.pre f).app X))
(CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.ev B).app X) h) = CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom f (CategoryTheory.CategoryStruct.id ((CategoryTheory.ihom A).obj X)))
(CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.ev A).app X) h)
@[simp]
theorem
CategoryTheory.MonoidalClosed.id_tensor_pre_app_comp_ev
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{B : C}
[CategoryTheory.Closed A]
[CategoryTheory.Closed B]
(f : B ⟶ A)
(X : C)
:
CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.CategoryStruct.id B)
((CategoryTheory.MonoidalClosed.pre f).app X))
((CategoryTheory.ihom.ev B).app X) = CategoryTheory.CategoryStruct.comp
(CategoryTheory.MonoidalCategory.tensorHom f (CategoryTheory.CategoryStruct.id ((CategoryTheory.ihom A).obj X)))
((CategoryTheory.ihom.ev A).app X)
@[simp]
theorem
CategoryTheory.MonoidalClosed.uncurry_pre
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{B : C}
[CategoryTheory.Closed A]
[CategoryTheory.Closed B]
(f : B ⟶ A)
(X : C)
:
@[simp]
theorem
CategoryTheory.MonoidalClosed.coev_app_comp_pre_app_assoc
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{B : C}
(X : C)
[CategoryTheory.Closed A]
[CategoryTheory.Closed B]
(f : B ⟶ A)
{Z : C}
(h : (CategoryTheory.ihom B).obj (CategoryTheory.MonoidalCategory.tensorObj A X) ⟶ Z)
:
CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app X)
(CategoryTheory.CategoryStruct.comp
((CategoryTheory.MonoidalClosed.pre f).app (CategoryTheory.MonoidalCategory.tensorObj A X)) h) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev B).app X)
(CategoryTheory.CategoryStruct.comp
((CategoryTheory.ihom B).map (CategoryTheory.MonoidalCategory.tensorHom f (CategoryTheory.CategoryStruct.id X)))
h)
@[simp]
theorem
CategoryTheory.MonoidalClosed.coev_app_comp_pre_app
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A : C}
{B : C}
(X : C)
[CategoryTheory.Closed A]
[CategoryTheory.Closed B]
(f : B ⟶ A)
:
CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app X)
((CategoryTheory.MonoidalClosed.pre f).app (CategoryTheory.MonoidalCategory.tensorObj A X)) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev B).app X)
((CategoryTheory.ihom B).map (CategoryTheory.MonoidalCategory.tensorHom f (CategoryTheory.CategoryStruct.id X)))
@[simp]
@[simp]
theorem
CategoryTheory.MonoidalClosed.pre_map
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{A₁ : C}
{A₂ : C}
{A₃ : C}
[CategoryTheory.Closed A₁]
[CategoryTheory.Closed A₂]
[CategoryTheory.Closed A₃]
(f : A₁ ⟶ A₂)
(g : A₂ ⟶ A₃)
:
theorem
CategoryTheory.MonoidalClosed.pre_comm_ihom_map
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{W : C}
{X : C}
{Y : C}
{Z : C}
[CategoryTheory.Closed W]
[CategoryTheory.Closed X]
(f : W ⟶ X)
(g : Y ⟶ Z)
:
CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoidalClosed.pre f).app Y) ((CategoryTheory.ihom W).map g) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom X).map g) ((CategoryTheory.MonoidalClosed.pre f).app Z)
@[simp]
theorem
CategoryTheory.MonoidalClosed.internalHom_obj
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
[CategoryTheory.MonoidalClosed C]
(X : Cᵒᵖ)
:
CategoryTheory.MonoidalClosed.internalHom.obj X = CategoryTheory.ihom X.unop
@[simp]
theorem
CategoryTheory.MonoidalClosed.internalHom_map
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
[CategoryTheory.MonoidalClosed C]
:
∀ {X Y : Cᵒᵖ} (f : X ⟶ Y), CategoryTheory.MonoidalClosed.internalHom.map f = CategoryTheory.MonoidalClosed.pre f.unop
def
CategoryTheory.MonoidalClosed.internalHom
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
[CategoryTheory.MonoidalClosed C]
:
The internal hom functor given by the monoidal closed structure.
Instances For
noncomputable def
CategoryTheory.MonoidalClosed.ofEquiv
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{D : Type u₂}
[CategoryTheory.Category.{v₂, u₂} D]
[CategoryTheory.MonoidalCategory D]
(F : CategoryTheory.MonoidalFunctor C D)
[CategoryTheory.IsEquivalence F.toFunctor]
[h : CategoryTheory.MonoidalClosed D]
:
Transport the property of being monoidal closed across a monoidal equivalence of categories
Instances For
theorem
CategoryTheory.MonoidalClosed.ofEquiv_curry_def
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{D : Type u₂}
[CategoryTheory.Category.{v₂, u₂} D]
[CategoryTheory.MonoidalCategory D]
(F : CategoryTheory.MonoidalFunctor C D)
[CategoryTheory.IsEquivalence F.toFunctor]
[CategoryTheory.MonoidalClosed D]
{X : C}
{Y : C}
{Z : C}
(f : CategoryTheory.MonoidalCategory.tensorObj X Y ⟶ Z)
:
CategoryTheory.MonoidalClosed.curry f = ↑(CategoryTheory.Adjunction.homEquiv (CategoryTheory.Functor.adjunction F.toFunctor) Y
((CategoryTheory.ihom (F.obj X)).obj
((CategoryTheory.Functor.inv (CategoryTheory.Functor.inv F.toFunctor)).obj Z)))
(CategoryTheory.MonoidalClosed.curry
(↑(CategoryTheory.Adjunction.homEquiv (CategoryTheory.Functor.adjunction (CategoryTheory.Functor.inv F.toFunctor))
(CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y)) Z)
(CategoryTheory.CategoryStruct.comp
((CategoryTheory.compInvIso (CategoryTheory.MonoidalFunctor.commTensorLeft F X)).hom.app Y) f)))
Suppose we have a monoidal equivalence F : C ≌ D, with D monoidal closed. We can pull the
monoidal closed instance back along the equivalence. For X, Y, Z : C, this lemma describes the
resulting currying map Hom(X ⊗ Y, Z) → Hom(Y, (X ⟶[C] Z)). (X ⟶[C] Z is defined to be
F⁻¹(F(X) ⟶[D] F(Z)), so currying in C is given by essentially conjugating currying in
D by F.)
theorem
CategoryTheory.MonoidalClosed.ofEquiv_uncurry_def
{C : Type u}
[CategoryTheory.Category.{v, u} C]
[CategoryTheory.MonoidalCategory C]
{D : Type u₂}
[CategoryTheory.Category.{v₂, u₂} D]
[CategoryTheory.MonoidalCategory D]
(F : CategoryTheory.MonoidalFunctor C D)
[CategoryTheory.IsEquivalence F.toFunctor]
[CategoryTheory.MonoidalClosed D]
{X : C}
{Y : C}
{Z : C}
(f : Y ⟶ (CategoryTheory.ihom X).obj Z)
:
CategoryTheory.MonoidalClosed.uncurry f = CategoryTheory.CategoryStruct.comp
((CategoryTheory.compInvIso (CategoryTheory.MonoidalFunctor.commTensorLeft F X)).inv.app Y)
(↑(CategoryTheory.Adjunction.homEquiv (CategoryTheory.Functor.adjunction (CategoryTheory.Functor.inv F.toFunctor))
(CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y)) Z).symm
(CategoryTheory.MonoidalClosed.uncurry
(↑(CategoryTheory.Adjunction.homEquiv (CategoryTheory.Functor.adjunction F.toFunctor) Y
((CategoryTheory.ihom (F.obj X)).obj (F.obj Z))).symm
f)))
Suppose we have a monoidal equivalence F : C ≌ D, with D monoidal closed. We can pull the
monoidal closed instance back along the equivalence. For X, Y, Z : C, this lemma describes the
resulting uncurrying map Hom(Y, (X ⟶[C] Z)) → Hom(X ⊗ Y ⟶ Z). (X ⟶[C] Z is
defined to be F⁻¹(F(X) ⟶[D] F(Z)), so uncurrying in C is given by essentially conjugating
uncurrying in D by F.)