A finitely presented $E_{\infty }$-prop I: algebraic context

Abstract

We introduce a finitely presented prop $\mathcal{S}=\{\mathcal{S}(n,m)\}$ in the category of differential graded modules whose associated operad $U(\mathcal{S})=\{\mathcal{S}(1,m)\}$ is a model for the $E_{\infty }$-operad. This finite presentation allows us to describe a natural $E_{\infty }$-coalgebra structure on the chains of any simplicial set in terms of only three maps: the Alexander-Whitney diagonal, the augmentation map, and an algebraic version of the join of simplices. The first appendix connects our construction to the Surjection operad of McClure-Smith and Berger-Fresse. The second establishes a duality between the join and AW maps for augmented and non-augmented simplicial sets. A follow up paper constructs a prop corresponding to $\mathcal{S}$ in the category of $CW$-complexes.

References in corpus (3)

• An axiomatic characterization of Steenrod's cup-$i$ products
• A finitely presented $E_{\infty }$-prop II: cellular context
• An algebraic representation of globular sets

Cited by in corpus (1)

• An algebraic representation of globular sets