# nLab accessible (infinity,1)-functor

Contents

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

This appears as HTT, def. 5.4.2.5.

## 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

Last revised on October 3, 2021 at 06:51:19. See the history of this page for a list of all contributions to it.