On sifted colimits in the presence of pullbacks

Abstract

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 $\kappa$-small sifted and filtered colimits, and their interaction with $\lambda$-small limits in place of finite ones, generalizing Garner's characterization of algebraic exactness in the sense of Ad\'amek--Lawvere--Rosick\'y. Along the way, we prove a general result on classes of colimits, showing that the $\kappa$-small restriction of a saturated class of colimits is still "closed under iteration".