# nLab accessible functor

Contents

category theory

## Applications

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## Definition

$F\colon C\to D$

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 $\kappa$.

## Properties

It is immediate from the definition that accessible functors are closed under composition.

### Raising the index of accessibility

If $\lambda\le\kappa$, then every $\kappa$-filtered colimit is also $\lambda$-filtered, and thus if $F$ preserves $\lambda$-filtered colimits then it also preserves $\kappa$-filtered ones. Therefore, if $F$ is $\lambda$-accessible and $C$ and $D$ are $\kappa$-accessible, then $F$ is $\kappa$-accessible. Two conditions under which this happens are:

1. $C$ and $D$ are locally presentable categories.

2. $\lambda$ is sharply smaller than $\kappa$, i.e. $\lambda\lhd\kappa$.

In particular, for any accessible functor $F$ there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible, and if the domain and codomain of $F$ are locally presentable then $F$ is $\kappa$-accessible for all sufficiently large $\kappa$.

### Preserving presentable objects

For any accessible functor $F$, there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible and preserves $\kappa$-presentable objects. Indeed, this can be achieved simultaneously for any set of accessible functors. See Adamek-Rosicky, Theorem 2.19.

### Essential images of accessible functors

###### Theorem

Assuming the existence of a proper class of strongly compact cardinals, the following are equivalent for the essential image $K$ of an accessible functor:

###### Theorem

Assuming the existence of a proper class of strongly compact cardinals, the closure of the image of an accessible functor under passage to subobjects is an accessible subcategory.

The existence of a proper class of strongly compact cardinals can be weakened, see the paper of Brooke-Taylor and Rosický.

## Examples

###### Example

Given locally presentable categories $C$ and $D$ and a functor $F\colon C\to D$, if $F$ has a left or right adjoint, then it is an accessible functor.

###### Example

By Example it follows that polynomial endofunctors of $Set$ are accessible, as they are composites of adjoint functors.

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.

Essential images of accessible functors are considered in

An improvement of Rosický’s result is in

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