mathlib documentation

category_theory.closed.cartesian

Cartesian closed categories #

Given a category with finite products, the cartesian monoidal structure is provided by the local instance monoidal_of_has_finite_products.

We define exponentiable objects to be closed objects with respect to this monoidal structure, i.e. (X × -) is a left adjoint.

We say a category is cartesian closed if every object is exponentiable (equivalently, that the category equipped with the cartesian monoidal structure is closed monoidal).

Show that exponential forms a difunctor and define the exponential comparison morphisms.

TODO #

Some of the results here are true more generally for closed objects and for closed monoidal categories, and these could be generalised.

@[reducible]

An object X is exponentiable if (X × -) is a left adjoint. We define this as being closed in the cartesian monoidal structure.

If X and Y are exponentiable then X ⨯ Y is. This isn't an instance because it's not usually how we want to construct exponentials, we'll usually prove all objects are exponential uniformly.

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The terminal object is always exponentiable. This isn't an instance because most of the time we'll prove cartesian closed for all objects at once, rather than just for this one.

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@[reducible]

A category C is cartesian closed if it has finite products and every object is exponentiable. We define this as monoidal_closed with respect to the cartesian monoidal structure.

@[reducible]

This is (-)^A.

Show that the exponential of the terminal object is isomorphic to itself, i.e. X^1 ≅ X.

The typeclass argument is explicit: any instance can be used.

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The internal hom functor given by the cartesian closed structure.

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If an initial object I exists in a CCC, then A ⨯ I ≅ I.

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If an initial object I exists in a CCC then it is a strict initial object, i.e. any morphism to I is an iso. This actually shows a slightly stronger version: any morphism to an initial object from an exponentiable object is an isomorphism.

If an initial object 0 exists in a CCC then every morphism from it is monic.