What are sifted colimits?
Sifted colimits play in algebraic categories the analogous role that filtered colimits do in finitely presentable categories. A category D is called sifted if D-colimits commute with finite products in Set. Algebraic categories can be characterized as free completions of small categoris under sifted colimits. And algebraic functors are precisely those preserving limits and sifted colimits.
Consequently, sifted colimits in algebraic categories are formed on the level of underlying sets.
We prove that sifted colimits are "essentially" just the combination
of filtered colimits and and reflexive coequalizers. For example, given a finitely cocomplete category A, then a functor with domain A preserves sifted colimits iff it preserves filtered colimits and reflexive coequalizers. However, for general categories A a counter-example
Jirí Adámek (TU Braunschweig, Germany)
November 10, 2009