= Rational polynomial differential forms
{title2=$A_{\mathrm{PL}}(X)$}
For the standard $n$-simplex use $\Omega_n=\mathbb Q[t_0,\ldots,t_n,dt_0,\ldots,dt_n]/(\sum t_i-1,\sum dt_i)$, with $|t_i|=0$, $|dt_i|=1$, and $d(t_i)=dt_i$. Compatible families of these forms on the simplices of a simplicial model of $X$ form the commutative <differential graded algebra> $A_{\mathrm{PL}}(X)$. Its <cohomology> is $H^*(X;\mathbb Q)$: polynomial contraction proves the simplex Poincaré lemma, and extension across faces gives the simplicial de Rham comparison. The contravariant functor takes simplicial pushouts to pullbacks of algebras.
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