On continuity of accessible functors

TL;DR

The paper establishes a criterion for the continuity of accessible functors between locally presentable categories, introduces a new adjoint functor theorem, and extends results to enriched categories, characterizing accessible small categories as Cauchy complete.

Contribution

It provides a new characterization of continuous accessible functors via preservation of small limits and extends the theory to enriched categories.

Findings

Accessible functors are continuous iff they preserve certain small limits.

A new adjoint functor theorem for accessible functors is derived.

Small enriched categories are accessible iff they are Cauchy complete.

Abstract

We prove that for each locally $\alpha$-presentable category $\mathcal K$ there exists a regular cardinal $\gamma$ such that any $\alpha$-accessible functor out of $\mathcal K$ (into another locally $\alpha$-presentable category) is continuous if and only if it preserves $\gamma$-small limits; as a consequence we obtain a new adjoint functor theorem specific to the $\alpha$-accessible functors out of $\mathcal K$. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small $\mathcal V$-category is accessible if and only if it is Cauchy complete.