2014/05/13 by Matěj Dostál, Dostál, Matěj, Jiřı́ Velebil +2
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.CT
paper · pdf · doi:10.48550/arxiv.1405.3090
An erroneous definition of saturated classes of weights was corrected on page 2
arxiv created 2014/06/23 · arxiv updated 2014/06/24
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.