# nLab accessible (infinity,1)-functor

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## 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

Revised on September 19, 2010 14:19:04 by Urs Schreiber (77.80.22.132)