# Is barycentric subdivision operator just a machinery to prove Excision theorem?

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.

• 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. – Danu Oct 6 '16 at 10:20
• 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 – user144221 Oct 6 '16 at 11:38
• 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. – Ronnie Brown Oct 6 '16 at 20:26

• 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.