Sufficient and necessary conditions for hereditarity of infinite category algebras

Abstract

We describe necessary and sufficient conditions for the hereditarity of the category algebra of an infinite EI category satisfying certain combinatorial assumptions. More generally, we discuss conditions such that the left global dimension of a category algebra equals the maximal left global dimension of the endomorphism algebras of its objects. As applications, we completely classify orbit categories, Quillen categories and transporter categories with left hereditary category algebras over a field.