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I am trying to read Goerss and Jardine's book (Simplicial Homotopy Theory) and in the proof of Theorem 7.10 (in chapter 1), they claim that there is a simplicial homotopy $\Delta^n\times\Delta^1\to\Delta^n$ given by the diagram:

$$\begin{array}{ccccccc} 0&\to &0&\to &\dots&\to &0\\ \downarrow &&\downarrow &&\dots &&\downarrow\\ 0&\to &1&\to&\dots&\to &n \end{array}$$ I am unsure how this actually defines a simplicial map at all. Is there a way to construct a simplicial map from this diagram?

If it helps, the simplicial map is (I think) supposed to be a contraction from the identity to 0.

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This simply contracts $\Delta^n$ onto the $0$-vertex. The diagram you draw encodes the following map $\mathbf{n} \times\mathbf{1} \to \mathbf{n}$ where elements $(m,0)\mapsto 0$ and $(m,1) \mapsto m$. This map induces $$ \Delta^n\times \Delta^1 = \operatorname{hom}_\Delta(-, \mathbf{n}) \times \operatorname{hom}_\Delta(-, \mathbf{1}) \cong \operatorname{hom}_\Delta(-,\mathbf{n} \times\mathbf{1}) \to \operatorname{hom}_\Delta(-, \mathbf{n}) = \Delta^n. $$ It could help if you check the effect after geometric realization. This is the standard contracting homotopy of $|\Delta^n|$ to $|\Delta^0| = *$. Write a formula for this homotopy and compare!

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  • $\begingroup$ Thank you for your answer! It makes the diagram very clear. $\endgroup$ – user722227 Apr 19 '20 at 12:33
  • $\begingroup$ Perhaps there is a typo in this books’s definition of homotopy, but wouldn’t this homotopy actually be from $0\to\Delta^n$? Certainly the restriction of this homotopy to $\Delta^n\times\{0\}$ is 0, right? At least this is true if the homotopy is defined by $(f,g)\to\varphi(f,g)$, where $\varphi$ is the map $\mathbf{n}\times\mathbf{1}\to\mathbf{n}$. $\endgroup$ – user722227 Apr 20 '20 at 17:47
  • $\begingroup$ I guess, but that seems irrelevant to me. $\endgroup$ – juan diego rojas Apr 21 '20 at 16:58
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Remember that a map between products of simplices is nothing more than a map of the corresponding partially ordered sets. Thus what is given here is the values of the desired map of the vertices of $\Delta^n\times \Delta^1$, which uniquely determine the map.

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