A. Hatcher here at pages 122-123 ha scritto: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 \(\Delta^n \to \Delta^n\).