
handle: 2078.1/94508
Sifted colimits play an important role in categorical algebra. Apart from their convenient objectwise construction in algebraic categories, they are used in the definition of strongly finitely presentable (or perfectly presentable) objects [see, e.g., \textit{J.~Adámek, J.~Rosický} and \textit{E.~M. Vitale}, Algebraic theories. A categorical introduction to general algebra. With a foreword by F. W. Lawvere. Cambridge Tracts in Mathematics 184. Cambridge: Cambridge University Press. (2011; Zbl 1209.18001)], which in a variety are just the retracts of free algebras on finitely many generators. This paper shows that every finitely cocomplete category has sifted colimits iff it has filtered colimits and reflexive coequalizers. Moreover, a functor with a finitely cocomplete domain preserves sifted colimits iff it preserves filtered colimits and reflexive coequalizers. Whereas the first statement is almost trivial, the proof of the second one is the main topic of the manuscript. The authors also show that both statements fail in case the assumption on finite cocompleteness is omitted.
sifted colimit, Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.), reflexive coequalizer, Categories admitting limits (complete categories), functors preserving limits, completions, Theories (e.g., algebraic theories), structure, and semantics, final functor, filtered colimit
sifted colimit, Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.), reflexive coequalizer, Categories admitting limits (complete categories), functors preserving limits, completions, Theories (e.g., algebraic theories), structure, and semantics, final functor, filtered colimit
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