I'm preparing a seminar on the barycentric subdivision of simplicial sets and I'm looking for some examples of this construction appearing in the literature. Since it's a useful technique (at least in the context of simplicial complexes), but it's very rarely the main subject of an article, it's hard for someone who's making its first steps in the area to find them. Would you mind sharing some?
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An important theoretical application is Kan's fibrant replacement functor $\def\Ex{{\sf Ex}}\def\Exi{\Ex^{\sf\infty}}\Exi$, defined as the filtered colimit of functors $\Ex^n$ ($n≥0$), where $\Ex$ is the right adjoint of the barycentric subdivision functor $\def\Sd{{\sf Sd}}\Sd$.
This construction is not just of theoretical interest, but also has practical applications, in particular, Kan's simplicial Whitehead theorem, which in its formulation for arbitrary simplicial sets requires a fibrant replacement functor, and by far the most practical choice of such a fibrant replacement is Kan's $\Exi$ functor.
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$\begingroup$ 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. $\endgroup$ Commented Aug 30, 2023 at 14:06
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1$\begingroup$ @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. $\endgroup$ Commented Aug 30, 2023 at 14:58