Geodesic
On sifted colimits in the presence of pullbacks
Theory and applications of categories, Tome 37 (2021), pp. 1176-1193.

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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".
Publié le :
Classification : 18A30, 18C35, 18E08
Keywords: sifted colimit, reflexive coequalizer, exact category, free cocompletion 
@article{TAC_2021_37_a34,
     author = {Ruiyuan Chen},
     title = {On sifted colimits in the presence of pullbacks},
     journal = {Theory and applications of categories},
     pages = {1176--1193},
     publisher = {mathdoc},
     volume = {37},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2021_37_a34/}
}
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Ruiyuan Chen. On sifted colimits in the presence of pullbacks. Theory and applications of categories, Tome 37 (2021), pp. 1176-1193. http://geodesic.mathdoc.fr/item/TAC_2021_37_a34/