A finitely presented ${E}_{\infty}$-prop II: cellular context

TL;DR

This paper constructs cellular models for $E_$-props using CW-complexes, enabling explicit cellular $E_$-bialgebra and coalgebra structures that connect to known operad models and prove conjectures.

Contribution

It introduces finitely generated cellular models for $E_$-props, providing explicit constructions of $E_$-bialgebra and coalgebra structures in a cellular context.

Findings

Constructed cellular $E_$-bialgebra structure on the interval.

Derived cellular $E_$-coalgebra structures on simplicial sets.

Connected constructions to Barratt-Eccles and Surjection coalgebras, proving a conjecture of Kaufmann.

Abstract

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.