An elementary characterisation of sifted weights

Abstract

Sifted colimits (those that commute with finite products in sets) play a major role in categorical universal algebra. For example, varieties of (many-sorted) algebras are precisely the free cocompletions under sifted colimits of (many-sorted) Lawvere theories. Such a characterisation does not depend on the existence of finite products in algebraic theories, but on the above fact that these products commute with sifted colimits and another condition: finite products form a sound class of limits. In this paper we study the notion of soundness for general classes of weights in enriched category theory. We show that soundness of a given class of weights is equivalent to having a `nice' characterisation of flat weights for that class. As an application, we give an elementary characterisation of sifted weights for the enrichment in categories and in preorders. We also provide a number of examples of sifted weights using our elementary criterion.