Given categories and , the functor category – written or – 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 has limits or colimits of a certain shape, then so does and they are computed pointwise. (However, if is not complete, then other limits in can exist “by accident” without being pointwise.)
If is small and is cartesian closed and complete, then is cartesian closed. See cartesian closed category for a proof.
If and are small, then is also small.
If is small and is locally small, then is still locally small.
Even if and are locally small, if is not small, then will usually not be locally small.
As a partial converse to the above, if and are locally small, then must be essentially small; see Freyd & Street (1995).
- (∞.1)-category of (∞,1)-functors?
Revised on September 21, 2012 10:12:56
by Urs Schreiber