Multisimplicial chains and configuration spaces

We define an \(E_\infty \) $E_{\infty }$ -coalgebra structure on the chains of multisimplicial sets. Our primary focus is on the surjection chain complexes of McClure-Smith, for which we construct a zig-zag of complexity preserving quasi-isomorphisms of \(E_\infty \) $E_{\infty }$ -coalgebras relating them to both the singular chains on configuration spaces and the Barratt–Eccles chain complexes.