nLab accessible (infinity,1)-functor

Idea

The generalization of the notion of accessible functor from category theory to (∞,1)-category theory.

Definition

Definition

An (∞,1)-functor $F \;\colon\; C \to D$ is accessible if $C$ is an accessible (∞,1)-category and there is a regular cardinal $\kappa$ such that $F$ preserves $\kappa$-filtered$\,$$(\infty,1)$-colimits.

This appears as HTT, def. 5.4.2.5 (and HTT, def. 5.3.4.5).

Properties

Proposition

(adjoint $(\infty,1)$-functors are accessible)
If an $(\infty,1)$-functor between accessible (∞,1)-categories has a left or right adjoint (∞,1)-functor, then it is itself accessible.

(HTT, prop. 5.4.7.7)

References

• Jacob Lurie, Section 5.4.2 of: Higher Topos Theory (2009)