Summary: Theory and Applications of Categories, Vol. 8, No. 3, , pp. 33--53.
ON SIFTED COLIMITS AND GENERALIZED VARIETIES
J. AD '
AMEK AND J. ROSICK '
Y
ABSTRACT. Filtered colimits, i.e., colimits over schemes D such that D­colimits in
Set commute with finite limits, have a natural generalization to sifted colimits: these
are colimits over schemes D such that D­colimits in Set commute with finite prod­
ucts. An important example: reflexive coequalizers are sifted colimits. Generalized
varieties are defined as free completions of small categories under sifted­colimits (anal­
ogously to finitely accessible categories which are free filtered­colimit completions of
small categories). Among complete categories, generalized varieties are precisely the va­
rieties. Further examples: category of fields, category of linearly ordered sets, category
of nonempty sets.
Introduction
Filtered colimits belong, no doubt, to the most basic concepts of category theory. Let us
just recall the notion of a finitely presentable object as one whose hom­functor preserves
filtered colimits. (This, in every variety of algebras, is equivalent to the usual -- less elegant
-- algebraic definition.)
Now, filtered colimits are characterized as colimits with domains (or diagram schemes)

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

Collections: Computer Technologies and Information Sciences