How accessible are categories of algebras?
The aim of the paper is to study properties of the category \({\mathcal A}lg F\) of \(F\)-algebras for a given endofunctor \(F:{\mathcal A}\to{\mathcal A}\). More precisely, let \(\lambda\) be a regular cardinal and \(\lambda^+\) its successor. An endofunctor \(F:{\mathcal A}\to{\mathcal A}\) is called \(\lambda\)-accessible (resp. strongly \(\lambda\)-accessible) provided that the category \({\mathcal A}\) is \(\lambda\)-accessible and \(F\) preserves \(\lambda\)-filtered colimits (resp. and \(\lambda\)-presentable objects). The following results are proved: if \({\mathcal A}\) is locally \(\lambda\)-multipresentable and \(F\) is \(\lambda\)-accessible, then \({\mathcal A}lg F\) is locally \(\lambda\)-multipresentable; if \(F\) is strongly \(\lambda\)-accessible, then \({\mathcal A}lg F\) is \(\lambda^+\)-accessible. Several examples and counterexamples are given.
scientific article; zbMATH DE number 1930188