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 '
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.
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)