On Sifted Colimits and Generalized Varieties

Filtered colimits, i.e., colimits over schemes D such that D-colimits in Set commute with nite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that D-colimits in Set commute with nite products. An important example: reeexive coequalizers are sifted colimits. Generalized varieties are deened as free completions of small categories under sifted-colimits (analogously to nitely accessible categories which are free ltered-colimit completions of small categories). Among complete categories, generalized varieties are precisely the varieties. Further examples: category of elds, category of linearly ordered sets, category of nonempty sets.