We show that in a category with pullbacks, arbitrary sifted colimits may be constructed as filtered colimits of reflexive coequalizers. This implies that "lex sifted colimits", in the sense of Garner-Lack, decompose as Barr-exactness plus filtered colimits commuting with finite limits. We also prove generalizations of these results for κ-small sifted and filtered colimits, and their interaction with λ-small limits in place of finite ones, generalizing Garner's characterization of algebraic exactness in the sense of Adámek-Lawvere-Rosický. Along the way, we prove a general result on classes of colimits, showing that the κ-small restriction of a saturated class of colimits is still "closed under iteration".
Keywords: sifted colimit, reflexive coequalizer, exact category, free cocompletion
2020 MSC: 18A30, 18C35, 18E08
Theory and Applications of Categories, Vol. 37, 2021, No. 35, pp 1176-1193.