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? $\endgroup$ Mar 28, 2017 at 21:42
  • $\begingroup$ @DenisNardin Yes I have modified the title. $\endgroup$ Mar 28, 2017 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 $\Delta_{p+1}$. $\endgroup$ Mar 29, 2017 at 5:18

2 Answers 2


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. $\endgroup$ Mar 29, 2017 at 7:45

The accepted answer is helpful but the proof in May's book is very terse; the combinatorics are unmotivated. I wanted to suggest a proof that is more enlightening.

The category of simplicial sets, like any presheaf category, is Cartesian closed. Given functors $A, B, C$ on a category $\mathcal{C}$, we want to define $C^B$ so that $Nat(A\times B,C)\cong Nat(A,C^B)$. Setting $A$ representable, say $y(c)$, we see that if such a functor $C^B$ exists, its value must be given on objects by Yoneda $C^B(c) \cong Nat(y(c),C^B) \cong Nat(y(c)\times B,C)$. Indeed, taking the definition of $C^B = Nat(y(-)\times B,C)$ gives a presheaf which can be checked to satisfy the exponential law.

In SSet we thus have that a homotopy $X\times I\to Y$ can be identified with a map from $X$ into the "path space" $Y^I$, where by the definition above, $Y^I_n = Nat(\Delta^1 \times \Delta^n,Y)$.

If we investigate the structure of $\Delta^1\times \Delta^n$, we see that all simplices in $(\Delta^1\times \Delta^n)_k$ are degenerate for $k>n+1$, and there are exactly $n+1$ simplices which are both nondegerate and not a face of any nondegenerate simplex; these are the $n+1$ nondegenerate simplices of $(\Delta^n\times \Delta^n)_{n+1}$, the injective maps $[n+1]\to [n]\times [1]$ (where the latter is equipped with the product ordering.) Every simplex in $\Delta^1\times \Delta^n$ can be expressed in terms of one of these principal simplices by repeatedly applying faces and degeneracies. So a natural transformation $\Delta^1\times \Delta^n\to Y$ is completely determined by where it sends the $n+1$ principal $n+1$-simplices, i.e. by a certain family of $n+1$ elements of $Y_{n+1}$.

A natural transformation $X\to Y^I$ can thus be expressed as a family of maps $h^q : X_q \to \prod_{0\leq i\leq q}Y_{q+1}$ where each $h^q(x) = \left\{h^q_i(x)\right\}$ codes the data of the natural transformation.

The conditions in May's book express the joint conditions that $h : X\to Y^I$ is natural and that each $h^q(x)$ actually defines a natural transformation $\Delta^1\times\Delta^n\to Y$.


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