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

This appears as HTT, def. 5.4.2.5.

Properties

Definition

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

This is HTT, prop. 5.4.7.7.

References

Section 5.4.2 of

• Jacob Lurie, Higher Topos Theory .