A Finitely Presented E-infinity Prop I

Anibal M. Medina-Mardones

Abstract


We introduce a finitely presented prop S={S(n,m)} in the category of differential graded modules whose associated operad U(S)={S(1,m)} is a model for the E-operad. This finite presentation allows us to describe a natural E-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 constructs a prop corresponding to S in the category of CW-complexes.

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