Let $f,g: X_{\bullet}\to Y_{\bullet}$ be two simplicial maps between simplicial sets. We say $f$ and $g$ are (strictly) simplicial homotopic if there exists a simplicial map $H: X_{\bullet}\times I_{\bullet}\to Y_{\bullet}$ such that $$ f=H\circ\varepsilon_0 \text{ and } g=H\circ\varepsilon_1 $$ where $\varepsilon_{\mu}: X_{\bullet}\to X_{\bullet}\times I_{\bullet}$, $\mu=0,1$ are the two obvious inclusions.

We also know that there is a combinatorial definition of simplicial homotopy between simplicial maps: We say $f$ and $g$ are (strictly) simplicial homotopic if if for each $p\geq 0$, there exists morphisms $$ h_i=h^p_i:X_p\to Y_{p+1} \text{ for } i=0,\ldots,p $$ such that the following conditions hold.

  1. $$\partial_0 h_0=f_p, \partial_{p+1}h_p=g_p;$$
  2. $$ \partial_ih_j=\begin{cases}h_{j-1}\partial_i & i<j\\ \partial_ih_{i-1} &i=j\\ h_j\partial_{i-1} & i>j+1 \end{cases};$$

  3. $$ s_ih_j=\begin{cases}h_{j+1}\partial_i & i\leq j\\ h_js_{i-1} & i>j \end{cases}. $$

My question is: could we find in the literature that these two versions of simplicial homotopy are equivalent?

  • $\begingroup$ I assume you want a reference and not just the proof? $\endgroup$ – Denis Nardin Mar 28 '17 at 21:42
  • $\begingroup$ @DenisNardin Yes I have modified the title. $\endgroup$ – Zhaoting Wei Mar 28 '17 at 22:28
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    $\begingroup$ The answer being there, the correspondence is very simple: a $p$-simplex $\sigma:\Delta_p\to X_\bullet$ gives $\sigma\times\text{identity}:\Delta_p\times I_\bullet\to X_\bullet\times I_\bullet$; the prism $\Delta_p\times I_\bullet$ consists of properly matched $p+1$ copies of $\Delta_{p+1}$ ($i$th facets of the ($i-1$)st and $i$th copy coincide for each $i=1,...,p+1$). Thus naming a map $X_\bullet\times I_\bullet\to Y_\bullet$ is the same as assigning to each $\sigma\in X_p$ a $p+1$-tuple of $p+1$-simplices in $Y_\bullet$ matching in the same way as in the above copies of $\Delta_{p+1}$. $\endgroup$ – მამუკა ჯიბლაძე Mar 29 '17 at 5:18

Proposition 6.2 in Chapter 1 of "Simplicial objects in algebraic topology", by J.P. May.

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    $\begingroup$ Theorem 8.3.2 of An introduction to homological algebra by C. Weibel, as well. $\endgroup$ – Andrea Gagna Mar 29 '17 at 7:45

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