2021/09/26 by Ruiyuan Chen, Chen, Ruiyuan
Mathematics · #18A30 #18C35 #18E08 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2109.12708
openalex publication_date 2021/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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".