functor category

Functor categories


Given categories CC and DD, the functor category – written D CD^C or [C,D][C,D] – is the category whose


Functor categories serve as the hom-categories in the strict 2-category Cat.

In the context of enriched category theory the functor category is generalized to the enriched functor category.

In the absence of the axiom of choice (including many internal situations), the appropriate notion to use is often instead the anafunctor category.


  • If DD has limits or colimits of a certain shape, then so does [C,D][C,D] and they are computed pointwise. (However, if DD is not complete, then other limits in [C,D][C,D] can exist “by accident” without being pointwise.)

  • If CC is small and DD is cartesian closed and complete, then [C,D][C,D] is cartesian closed. See cartesian closed category for a proof.

Size issues

If CC and DD are small, then [C,D][C,D] is also small.

If CC is small and DD is locally small, then [C,D][C,D] is still locally small.

Even if CC and DD are locally small, if CC is not small, then [C,D][C,D] will usually not be locally small.

As a partial converse to the above, if CC and [C,Set][C,Set] are locally small, then CC must be essentially small; see Freyd & Street (1995).

  • (∞.1)-category of (∞,1)-functors?

Revised on September 21, 2012 10:12:56 by Urs Schreiber (