An elementary characterisation of sifted weights

TL;DR

This paper provides an elementary and general characterization of sifted weights in enriched category theory, linking soundness and flatness, with applications to categories and preorders.

Contribution

It introduces a new elementary criterion for sifted weights, connecting soundness with flatness, and applies it to categories and preorders.

Findings

Elementary characterization of sifted weights in enriched categories

Equivalence between soundness and flatness of weights

Multiple examples illustrating the criterion

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.