  We construct, using finitely many generating cell and relations, props in the
category of CW-complexes with the property that their associated operads are
models for the $E_\infty$-operad. We use one of these to construct a cellular
$E_\infty$-bialgebra structure on the interval and derive from it a natural
cellular $E_\infty$-coalgebra structure on the geometric realization of a
simplicial set which, passing to cellular chains, recovers up to signs the
Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse
and McClure-Smith. We use another prop, a quotient of the first, to relate our
constructions to earlier work of Kaufmann and prove a conjecture of his. This
is the second of two papers in a series, the first investigates analogue
constructions in the category of chain complexes.
