In ordinary categories both filtered colimits and reflexive coequalizers have the property that in every variety of algebras such colimits are formed on the level of the underlying sets. More generally, all sifted colimits have that property. A category \( \mathcal D \) is sifted if colimits of diagrams over \(\mathcal{D}\) in \(\mathbf{Set}\) commute with finite products. It turns out that sifted colimits are essentially just filtered colimits combined with reflexive coequalizers.
Analogously, in \(\mathcal{V}\)categories, sifted colimits are those weighted by sifted weights \(W: \mathcal{D ^{op}} \to \mathcal{V}\). This means that colimits weighted by \(W\) commute in \(\mathcal{V}\) with finite products. For example, in the cartesian closed category of posets they are essentially just combinations of filtered colimits and reflexive coinserters.
Sifted colimits play a central role in varieties: every variety is the free completion of \(T^{op}\) (where \(T\) is its algebraic theory) under sifted colimits.
