# Where to find the proof that these two version of simplicial homotopy are equivalent?

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?

• I assume you want a reference and not just the proof? Mar 28, 2017 at 21:42
• @DenisNardin Yes I have modified the title. Mar 28, 2017 at 22:28
• 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 $\Delta_{p+1}$. Mar 29, 2017 at 5:18

## 2 Answers

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

• Theorem 8.3.2 of An introduction to homological algebra by C. Weibel, as well. 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 $$\Delta^1\times\Delta^n\to Y$$.