Cartesian Closed Category

A Category $\mathcal{C}$ is cartesian closed if it is

• A Cartesian Category
• with all Exponential Objects

Global Definition

There is also a nice global definition: a category $\mathcal{C}$ is cartesian closed if and only if for every $A$, the functor $A\times-:\mathcal{C}\to\mathcal{C}$ has a Right Adjoint. Note that we do not need an Extranaturality condition on $A$; the universal property takes care of that for us.

Via Monoidal Closure

A category $\mathcal{C}$ is cartesian closed if it is a Closed Monoidal Category with respect to the cartesian monoidal structure.

Examples

• The canonical example of a cartesian closed category is the Category of Sets.

Nonexamples

• The Category of Topological Spaces is notably not cartesian closed.