A functor
is a -accessible functor (for a regular cardinal) if and are both -accessible categories and preserves -filtered colimits. is an accessible functor if it is -accessible for some regular cardinal .
If , then every -filtered colimit is also -filtered, and thus if preserves -filtered colimits then it also preserves -filtered ones. Therefore, if is -accessible and and are -accessible, then is -accessible. Two conditions under which this happens are:
and are locally presentable categories.
is sharply smaller than , i.e. .
In particular, for any accessible functor there are arbitrarily large cardinals such that is -accessible, and if the domain and codomain of are locally presentable then is -accessible for all sufficiently large .
For any accessible functor , there are arbitrarily large cardinals such that is -accessible and preserves -presentable objects. Indeed, this can be achieved simultaneously for any set of accessible functors. See Adamek-Rosicky, Theorem 2.19.
The theory of accessible 1-categories is described in
The theory of accessible -categories is the topic of section 5.4 of
Last revised on March 1, 2019 at 18:59:43. See the history of this page for a list of all contributions to it.