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Let $S_X$ be the barycentric subdivision operator of a topological space $X$ in singular theory. (The one standard algebraic topology texts such as Hatcher and Munkres) There is also a machinery so called barycentric subdivision in simplicial theory. The idea of constructions are the same, but they are different. I wonder if there is a reason not to distinguish them in terminologies.

So, my question is whether $S_X$ is just a machinery to prove Excision theorem. I skimmed my texts, but I clould not find $S_X$ after the proof of excision theorem.

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  • $\begingroup$ I'm not an expert, but I'm pretty sure the answer is yes. I never saw the operator show up except in setting up the excision theorem. $\endgroup$ – Danu Oct 6 '16 at 10:20
  • $\begingroup$ This "operator" as seen in Hatcher's text is indeed a rather arbitrary thing. To prove the excision, you need to subdivide simplices into smaller simplices, and that "operator" gives you one canonical procedure. mathoverflow.net/questions/31035 $\endgroup$ – user144221 Oct 6 '16 at 11:38
  • $\begingroup$ I'm fond of the cubical approach in Sch\"on, R. , "Acyclic models and excision", Proc. Amer. Math. Soc. 59~(1) (1976) 167--168, as the "standard" subdivision of a cube into small cubes is very easy to picture, and can be exploited in other ways. $\endgroup$ – Ronnie Brown Oct 6 '16 at 20:26
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Barycentric subdivision, or rather subdivision of simplicial sets, does indeed show up elsewhere in algebraic topology. See the answers from the link in Alejo's comment. Here are a couple more:

  • The subdivision functor has a right adjoint $\operatorname{Ex}$, which is used to construct the Kan fibrant replacement functor $\operatorname{Ex}^\infty$ in the model structure on simplicial sets.

  • "The second subdivision of a simplicial set is a simplicial complex." "The second subdivision of a category is a poset." See the second part of May's book-in-progess on finite spaces for an explanation of these statements.

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