On continuity of accessible functors

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.

Revised version, minor typos fixed, published on Applied Categorical Structures

References in corpus (1)

• Flat vs. filtered colimits in the enriched context

Cited by in corpus (1)

• Accessible categories with a class of limits