nLab
accessible functor

Definition

For categories

A functor

$$F:C\to D$$F\colon C\to D

is a $\kappa$-accessible functor (for $\kappa$ a regular cardinal) if $C$ and $D$ are both $\kappa$-accessible categories and $F$ preserves $\kappa$-filtered colimits. $F$ is an accessible functor if it is $\kappa$-accessible for some regular cardinal $\kappa$.

Higher categorical version

See accessible (∞,1)-functor.

References

The theory of accessible 1-categories is described in

• Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.1989.

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

The theory of accessible $(\mathrm{\infty },1)$-categories is the topic of section 5.4 of

• Jacob Lurie, Higher Topos Theory