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Theory and Applications of Categories
Article . 2010 . Peer-reviewed
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zbMATH Open
Article . 2010
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What are Sifted Colimits?

What are sifted colimits?
Authors: Adamek, J.; Rosicky, Jiri; Vitale, Enrico;

What are Sifted Colimits?

Abstract

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.

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Belgium
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Keywords

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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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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