Filtered colimits, i.e., colimits over schemes $\cal D$ such that $\cal D$-colimits in $\Set$ commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes $\cal D$ such that $\cal D$-colimits in $\Set$ commute with finite products. An important example: reflexive coequalizers are sifted colimits. Generalized varieties are defined as free completions of small categories under sifted-colimits (analogously to finitely accessible categories which are free filtered-colimit completions of small categories). Among complete categories, generalized varieties are precisely the varieties. Further examples: category of fields, category of linearly ordered sets, category of nonempty sets.
Theory and Applications of Categories, Vol. 8, 2001, No. 3, pp 33-53.
Revised 2007-01-22. Original version at