On sifted colimits in the presence of pullbacks

TL;DR

This paper characterizes how sifted colimits can be constructed from filtered colimits and reflexive coequalizers in categories with pullbacks, extending known results to small colimits and their interaction with limits.

Contribution

It provides a new decomposition of lex sifted colimits and generalizes algebraic exactness to small colimits and limits, with a focus on classes of colimits.

Findings

Sifted colimits can be expressed as filtered colimits of reflexive coequalizers.

Lex sifted colimits decompose into Barr-exactness plus filtered colimits commuting with finite limits.

Generalization of algebraic exactness for small colimits and limits.

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".