Simplicial set map from $\Delta[n]$ to $K$ induced by an $n$-simplex $\sigma \in K_n.$

Question

I am reading "rational homotopy theory" by Yves Felix, Stephen Halperin and Jean-Claude Thomas on simplicial sets and simplicial cochain algebras. There is a lemma 10.3 page 118 which states that

If $K$ is any simplicial set then any $\sigma\in K_n$ determines a unique simplicial set map $\sigma_*:\Delta[n]\to K$ such that $\sigma_*(c_n)=\sigma$.

Here $c_n$ is the fundamental simplex of $\Delta[n]$.

What I tried: In order to get a map $\sigma_*:\Delta[n]\to K$, we need to construct a sequence of maps $(\sigma_*)_k: \Delta[n]_k \to K_k$ for all $k\geq 0$ such that $ (\sigma_*)_k \circ\tilde{\partial}_i^{k+1} = \partial_i^{k+1} \circ (\sigma_*)_{k+1}$ and $s_i^{k+1}\circ(\sigma_*)_k = (\sigma_*)_{k+1}\circ\tilde{s}_i^{k+1} $ for all $k\geq 0$, where $\tilde{\partial}_i^k , \partial_i^k $ are face maps, $\tilde{s}_i^k, s_i^k $ are degeneracy maps of $\Delta[n]$ and $K$ respectively, for all $k\geq 0$. That is the following two ladders are commutative: $\require{AMScd}$ \begin{CD} \Delta[n] : \cdots @>>>\Delta[n]_n @>{\tilde{\partial}_i^n}>> \cdots @>>> \Delta[n]_{k+1} @>{\widetilde{\partial}^{k+1}_i}>> \Delta[n]_k @>{\widetilde{\partial}^{k}_i}>> \cdots @>>> \Delta[n]_1 @>{\widetilde{\partial}^1_i}>>\Delta[n]_0 \\ @V{\sigma_*}VV@V{(\sigma_*)_n}VV@.@V{(\sigma_*)_{k+1}}VV@V{(\sigma_*)_k}VV@.@V{(\sigma_*)_1}VV@V{(\sigma_*)_0}VV\\ K: \cdots @>>> K_n @>{\partial^n_i}>>\cdots @>>>K_{k+1} @>{\partial^{k+1}_i}>> K_k @>{\partial^k_i}>> \cdots @>>> K_1 @>{\partial^1_i}>>K_0 \end{CD}

\begin{CD} \Delta[n] : \cdots @<<<\Delta[n]_n @<{\tilde{s}_i^n}<< \cdots @<<< \Delta[n]_{k+1} @<{\widetilde{s}^{k+1}_i}<< \Delta[n]_k @<{\widetilde{s}^{k}_i}<< \cdots @<<< \Delta[n]_1 @<{\widetilde{s}^1_i}<<\Delta[n]_0 \\ @V{\sigma_*}VV@V{(\sigma_*)_n}VV@.@V{(\sigma_*)_{k+1}}VV@V{(\sigma_*)_k}VV@.@V{(\sigma_*)_1}VV@V{(\sigma_*)_0}VV\\ K: \cdots @<<< K_n @<{s^n_i}<<\cdots @<<<K_{k+1} @<{s^{k+1}_i}<< K_k @<{s^k_i}<<\cdots @<<< K_1 @<{s^1_i}<<K_0 \end{CD}

First I define $(\sigma_*)_0:\Delta[n]_0 \to K_0$ by $(\sigma_*)_0(<e_j>) = \partial_0^1 \circ \cdots \circ \partial_0^n (\sigma)$, and for any $k>0$ define $(\sigma_*)_k:\Delta[n]_k\to K_k$ by $(\sigma_*)_k(<e_{j_0}, \ldots, e_{j_k}>) = s_i^k \circ (\sigma_*)_{k-1} \circ \tilde{\partial}_j^k(<e_{j_0}, \ldots, e_{j_k}>)$ for any choice of $i$ and $j$. I manage to show that the $(\sigma_*)_k$ defined as above are well defined irrespective of the choice of $i$ and $j$ in the definition and that both of the two ladders are commutative. That is $\sigma_*$ is a simplicial set map, but I cannot prove that $(\sigma_*)_n (c_n) =\sigma.$

Question: Is my definition of $\sigma_*$ correct? if this is not the right way to define $\sigma_*$, then how can I define? secondly if the definition of $\sigma_*$ is correct can we show that $(\sigma_*)_n(c_n)=\sigma$.

Any help is appreciated.

Answer 1

I have not checked your definition of $\sigma_*$ carefully, but I suspect it is not correct. (For example, your definition of $(\sigma_*)_0(\langle e_j \rangle)$ does not depend on $j$, which is a bit suspicious.) I will sketch a slightly different approach, and leave the details for you.

The key point is that any element $\alpha \in \Delta[n]_k$ can be written as $s_{j_0} \cdots s_{j_a} \partial_{\ell_0} \cdots \partial_{\ell_b}(c_n)$ for some sequence of $j$s and $\ell$s. The idea here is that, if $\alpha = \langle e_{i_0}, \ldots, e_{i_k} \rangle$, then we can use the $\partial$s to "get rid of any unused elements in the sequence $\langle e_0, \ldots, e_n \rangle$", and then use the $s$s to " introduce any necessary repetitions". I encourage you to work out a few examples to see what I mean.

Now, to define $(\sigma_*)_k$, we proceed as follows. For $\alpha \in \Delta[n]_k$, we write $\alpha = s_{j_0} \cdots s_{j_a} \partial_{\ell_0} \cdots \partial_{\ell_b}(c_n)$. Then $(\sigma_*)_k(\alpha) = s_{j_0} \cdots s_{j_a} \partial_{\ell_0} \cdots \partial_{\ell_b}(\sigma)$.

I will leave it to you to check all the necessary details for this definition. I will also mention that there is another (essentially equivalent) approach to the theory of simplicial sets (via presheaves on the simplex category) that makes this problem much easier to deal with.