I am currently reading Section 10 in Rational Homotopy Theory by Felix, Halperin and Thomas.

A rather important step after the construction of the simplicial cochain algebra $A = A_{PL}$ is proving they are extendable, meaning that for any (not necessarily strict) subset $I$ of $\{0,\ldots,n\}$ and simplices $\Phi_i$ for which $\partial_i\Phi_j = \partial_{j-1}\Phi_i$, $i<j$ in $I$, there is $\Phi$ such that $\partial_i \Phi = \Phi_i$.

This is claim $(iii)$ in Lemma 10.7, page 123 of the first edition.

Concretely, $A_n$ is the free commutative DG-algebra generated by $t_0,\ldots,t_n$ in degree $0$, by $y_0,\ldots,y_n$ in degree $1$ and with $dt_i = y_i$, modulo the relations that $\sum t_i = 1$ and $\sum y_i=0$, face/coface maps given by thinking about the $t_i$ as the barycentric coordinates of the $n$-simplex. Equivalently, it is generated by $t_1,\ldots,t_n$ and $y_1,\ldots,y_n$ subject to commutativity only and $t_i$ covers $y_i$.

I find the proof really obscure; although with a bit of work I can follow it, I cannot get anything out of it. I was wondering if someone could shed some light on the proof, or else point to other proofs (if there are any). I admit the whole book is rather terse at moments, but in general, independent of this, I can gather some intuition of what is going on, but this particular bit is eluding me.

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Don't know is this question still actual, but you can find an answer below.

First of all, you can find more geometric proof in Griffiths, Morgan -- Rational Homotopy Theory and Differential Forms.

In few words: Let $p\colon\Delta^n\to \Delta^{n-1} = \{t_0 = 0\}$ be a stereographic projection from the vertex $t_0 = 1$, and let $\alpha \in \Omega^*(\Delta^{n-1})$ be a polynomial form on the face $t_0 = 0$. Then $p^*(\alpha)$ is polynomial from from generators $t_0, \dots, t_{n-1}, \frac{1}{1-t_n}, dt_1, \dots, dt_n$. Therefore, $(1-t_n)^Np^*(\alpha)$ is actually polynomial form on $\Delta^n$.

Let $\varphi \in \Omega^*(\partial\Delta[n])$. We now build forms $\psi_j$, s.t. $\varphi - \sum\limits_{0}^k \psi_i$ vanishes on the union $\sigma_0\cup\dots\cup\sigma_k$, where $\sigma_i = d^i[n-1]$ is $(n-i)$-th face. Namely, let $\psi_0$ be an extension of $\varphi|_{\sigma_0}$ to the full simplex $\Delta[n]$. Now, let $\psi_1$ be an extension of $\varphi_1 = (\varphi - \psi_0)|_{\sigma_1}$. Then $\varphi - \psi_0 - \psi_1$ vanishes on $\sigma_0\cup\sigma_1$. Now, we continue inductively. Hence, we get $\varphi = \sum\limits_{i = 0}^n\psi_i|_{\partial\Delta[n]}$, proving that $\Omega^*$ is extandable.

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