# On Finitary Functors

## TL;DR

The paper introduces a criterion for identifying finitary functors based on finitely bounded subobjects, generalizes this to various categories, and applies it to show the Hausdorff functor is accessible.

## Contribution

It provides a simple criterion for finitary functors, extends the concept to locally presentable categories, and applies it to the Hausdorff functor.

## Key findings

- Finitely bounded functors are equivalent to finitary functors under certain conditions.
- The criterion applies to categories like Set, vector spaces, and boolean algebras.
- The Hausdorff functor on complete metric spaces is shown to be -accessible.

## Abstract

A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is equivalent to $F$ being finitary for all functors between `reasonable' locally finitely presentable categories, provided that $F$ preserves monomorphisms. We also discuss the question when that last assumption can be dropped. The answer is affirmative for functors between categories such as Set, K-Vec (vector spaces), boolean algebras, and actions of any finite group either on Set or on K-Vec for fields K of characteristic 0. All this generalizes to locally $\lambda$-presentable categories, $\lambda$-accessible functors and $\lambda$-presentable algebras. As an application we obtain an easy proof that the Hausdorff functor on the category of complete metric spaces is $\aleph_1$-accessible.