nLab accessible functor

Definition

A functor

$$F\colon C\to D$$

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:

• $K$ is complete;
• $K$ is cocomplete;
• $K$ is locally presentable.

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.

Related concepts

• accessible (∞,1)-functor.

• accessible monad

References

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.

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

Essential images of accessible functors are considered in

• Jiří Rosický, More on directed colimits of models, Applied Categorical Structures volume 2 (1994) pages 71–76, doi:10.1007/BF00878503

An improvement of Rosický’s result is in

• Andrew Brooke-Taylor, Jiří Rosický, Accessible images revisited (arXiv:1506.01986).

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

• Jacob Lurie, Higher Topos Theory