A recognition principle for iterated suspensions as coalgebras over the little cubes operad

2022/10/03 by Flynn-Connolly, Oisín, Moreno-Fernández, José M., Wierstra, Felix
#18M75 #55P40 #55P48 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.00839

Abstract

Our main result is a recognition principle for iterated suspensions as coalgebras over the little disks operads. Given a topological operad, we construct a comonad in pointed topological spaces endowed with the wedge product. We then prove an approximation theorem that shows that the comonad associated to the little n-cubes operad is weakly equivalent to the comonad Σⁿ Ωⁿ arising from the suspension-loop space adjunction. Finally, our recognition theorem states that every little n-cubes coalgebra is homotopy equivalent to an n-fold suspension. These results are the Eckmann--Hilton dual of May's foundational results on iterated loop spaces.

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