nLab
cartesian closed category

Definition

A cartesian closed category (sometimes: ccc) is a category with finite products which is closed with respect to its cartesian monoidal structure.

Examples

Inheritance by reflective subcategories

In showing that a given category is cartesian closed, the following theorem is often useful (cf. A4.3.1 in the Elephant):

Theorem

If C is cartesian closed, and DC is a reflective subcategory, then the reflector L:CD preserves finite products if and only if D is an exponential ideal (i.e. YD implies Y XD for any XC). In particular, if L preserves finite products, then D is cartesian closed.

Remarks