Documentation

Mathlib.CategoryTheory.Closed.Monoidal

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.

An object X is (right) closed if (X ⊗ -) is a left adjoint.

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    A monoidal category C is (right) monoidal closed if every object is (right) closed.

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      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.

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        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.

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          A ⟶[C] B denotes the internal hom from A to B

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            Currying in a monoidal closed category.

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              Uncurrying in a monoidal closed category.

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                Pre-compose an internal hom with an external hom.

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                  The internal hom functor given by the monoidal closed structure.

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                    Transport the property of being monoidal closed across a monoidal equivalence of categories

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                      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) {G : CategoryTheory.Functor D C} (adj : F.toFunctor G) [F.IsEquivalence] [CategoryTheory.MonoidalClosed D] {X : C} {Y : C} {Z : C} (f : CategoryTheory.MonoidalCategory.tensorObj X Y Z) :
                      CategoryTheory.MonoidalClosed.curry f = (adj.homEquiv Y ((CategoryTheory.ihom (F.obj X)).obj (F.obj Z))) (CategoryTheory.MonoidalClosed.curry ((adj.toEquivalence.symm.toAdjunction.homEquiv (CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y)) Z) (CategoryTheory.CategoryStruct.comp ((F.commTensorLeft X).compInverseIso.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) {G : CategoryTheory.Functor D C} (adj : F.toFunctor G) [F.IsEquivalence] [CategoryTheory.MonoidalClosed D] {X : C} {Y : C} {Z : C} (f : Y (CategoryTheory.ihom X).obj Z) :
                      CategoryTheory.MonoidalClosed.uncurry f = CategoryTheory.CategoryStruct.comp ((F.commTensorLeft X).compInverseIso.inv.app Y) ((adj.toEquivalence.symm.toAdjunction.homEquiv ((F.comp (CategoryTheory.MonoidalCategory.tensorLeft (F.obj X))).obj Y) Z).symm (CategoryTheory.MonoidalClosed.uncurry ((adj.homEquiv Y ((CategoryTheory.ihom (F.obj X)).obj (adj.toEquivalence.symm.inverse.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.)