nLab
functor category

Definition

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

• objects are functors $F:C\to D$

• morphisms are natural transformations between these functors.

Usage

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.

Properties

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

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

Size issues

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

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

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

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

Related concepts

• (∞.1)-category of (∞,1)-functors?