Constructing a simplicial map from a diagram

Question

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.

Answer 1

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!

Answer 2

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.

Answer 3

Here's another approach. Define the natural transformation $H:\Delta^n\times\Delta^1\twoheadrightarrow\Delta^n$ whose component at $\mathbf{m}$ is given by $$ \begin{array}{ccc} H_\mathbf{m}:\Delta^n(\mathbf{m})\times\Delta^1(\mathbf{m}) &\twoheadrightarrow& \Delta^n(\mathbf{m})\\ (\alpha,\beta)&\mapsto&\alpha\cdot\beta \end{array} $$ where $\alpha\cdot\beta:\mathbf{m}\rightarrow\mathbf{n}$ is the nondecreasing map given by pointwise multiplication. The natural transformation $H$ is a simplicial homotopy from the zero map $0:\Delta^n\rightarrow\Delta^n$ to the identity map $1_{\Delta^n}:\Delta^n\rightarrow\Delta^n$.