Timeline for Big list: barycentric subdivision of simplicial sets
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4 events
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| Aug 30, 2023 at 14:58 | comment | added | Dmitri Pavlov | @TimCampion: There is also an explicit description of maps $\def\Ex{\mathop{\sf Ex}}A→\Ex^∞ X$ for a simplicial set $A$ with finitely many simplices: these are precisely the maps $\def\Sd{\mathop{\sf Sd}}\Sd^k A→X$ for some $k≥0$, with an obvious equivalence relation. There is no such simplicial description for the singular complex functor. This is useful in practice, e.g., for the main theorem of arxiv.org/abs/2110.04679. | |
| Aug 30, 2023 at 14:06 | comment | added | Tim Campion | Note that many of the nice formal properties of $Ex^\infty$ -- e.g. preserving finite limits, fibrations, and weak equivalences while being a fibrant replacement functor -- are shared by the functor $X \mapsto Sing(|X|)$ -- the singular simplicial set of the geometric realization. One property which makes $Ex^\infty$ "better" is that it preserves filtered colimits. This is useful e.g. for showing that weak equivalences of simplicial sets are closed under filtered colmits. And of course, $Ex^\infty X$ is much smaller than $Sing(|X|)$, which is psychologically reassuring. | |
| May 20, 2023 at 3:07 | history | made wiki | Post Made Community Wiki by David Roberts♦ | ||
| May 18, 2023 at 21:02 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |