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Keywords:
Operads; Props; $E_\infty$-structures; Normalized chains
Operads; Props; $E_\infty$-structures; Normalized chains
Summary:
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 simplicial sets 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 diagonal and join maps for chains of augmented and non-augmented simplicial sets. A follow up paper [MM18b] constructs a prop corresponding to $\mathcal{S}$ in the category of $CW$-complexes.
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