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  1. (, Theory and applications of categories)
    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". 
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