Hatcher's barycentric subdivision of chains

[kaspar]

Hi, I am having trouble understanding the following part:

Barycentric Subdivision of General Chains. Define \(S : C_n(X) \to C_n(X)\) by setting \(S\sigma = \sigma_\sharp S\Delta^n\) for a singular \(n\)-simplex \(\sigma : \Delta^n \to X\). [...]

In particular: how shall I parse the piece \(S\Delta^n\)? By a quick type analysis, \(S\) expects a linear chain, so how do I have to interpret \(\Delta^n\)? In the calculations below it seems that it is used as an actual identity map \(\Delta^n \to \Delta^n\).

[megas_archon]

$S$ is not a random family of maps, it must be natural in $X$ (this means that it has to be a natural transformation of the functor \(C_n\)): so, by Yoneda lemma, it is completely determined by its action on the universal element of \(\Delta^n\), i.e. the identity simplex \(1 : \Delta^n \to \Delta^n \).

adiaŭ!

[kaspar]

Yes, there is some scent of such things. ^-^

Thanks.

[megas_archon]