In the context of differential graded modules, models of the \(E_\infty \)-operad are of central importance. Props generalise operads by allowing morphisms with not just a finite tuple of inputs but also a finite tuple of outputs. Consequentely, the combinatorics of props is much more intricate since arities can increase and decrease through composition. The author takes advantage of props by presenting a prop \(\mathcal S\) given by three generators, one of which is of signature \((1,2)\) and its differential vanishes. It is then shown that the associated operad \(\{\mathcal S(1,m)\}\) is a model for the \(E_\infty \)-operad (but \(\{\mathcal S(m,1)\}\) is not).
The finite presentation allows the author to define an \(S\)-bialgebra structure on the chains of standard simplices as well as an associated \(E_\infty \)-coalgbera structure on the chains of simplicial sets. These constructions form the main part of the article. The remaining two appendices demonstrate, respectively, that a suitable quotient \(\mathcal {MS}\) of the prop \(\mathcal S\) has as associated operad the Surjection operad \(\mathcal {S}ur\), and a certain duality in chains of simplicial sets.