How accessible are categories of algebras?

Abstract

For locally finitely presentable categories K it is well known that categories of F-algebras, where F is a finitary endofunctor, are also locally finitely presentable. We prove that this generalizes to locally finitely multipresentable categories. But it fails, in general, for finitely accessible categories: we even present an example of a strongly finitary functor F (one that preserves finitely presentable objects) whose category of F-algebras is not finitely accessible. On the other hand, categories of F-algebras are proved to be ω 1-accessible for all strongly finitary functors—and it is an open problem whether this holds for all finitary functors.