nLab
functor category
Context
Category theory
category theory

Concepts
Universal constructions
Theorems
Extensions
Applications
Functor categories
Definition
Given categories $C$ and $D$ , the functor category – written $D^C$ or $[C,D]$ – is the category whose

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 $[C,Set]$ are locally small, then $C$ must be essentially small ; see Freyd & Street (1995) .

(∞.1)-category of (∞,1)-functors?

Revised on September 21, 2012 10:12:56
by

Urs Schreiber
(82.169.65.155)