Rational polynomial differential forms ($A_{\mathrm{PL}}(X)$)

For the standard $n$-simplex use ${\Omega }_n=\mathbb{Q}[t_0,\dots ,t_n,d{t}_0,\dots ,d{t}_n]/(\sum t_i-1,\sum d{t}_i)$, with $|t_i|=0$, $|d{t}_i|=1$, and $d(t_i)=d{t}_i$. 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 $H^{\ast }(X;\mathbb{Q})$: 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.