nLab
accessible functor

Contents

Contents

Definition

A functor

F:CD F\colon C\to D

is a κ\kappa-accessible functor (for κ\kappa a regular cardinal) if CC and DD are both κ\kappa-accessible categories and FF preserves κ\kappa-filtered colimits. FF is an accessible functor if it is κ\kappa-accessible for some regular cardinal κ\kappa.

Properties

Raising the index of accessibility

If λκ\lambda\le\kappa, then every κ\kappa-filtered colimit is also λ\lambda-filtered, and thus if FF preserves λ\lambda-filtered colimits then it also preserves κ\kappa-filtered ones. Therefore, if FF is λ\lambda-accessible and CC and DD are κ\kappa-accessible, then FF is κ\kappa-accessible. Two conditions under which this happens are:

  1. CC and DD are locally presentable categories.

  2. λ\lambda is sharply smaller than κ\kappa, i.e. λκ\lambda\lhd\kappa.

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

Preserving presentable objects

For any accessible functor FF, there are arbitrarily large cardinals κ\kappa such that FF is κ\kappa-accessible and preserves κ\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 KK 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.

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.

Essential images of accessible functors are considered in

An improvement of Rosický’s result is in

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

Last revised on June 11, 2020 at 01:13:53. See the history of this page for a list of all contributions to it.