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Our main result is a recognition principle for iterated suspensions ascoalgebras over the little disks operads. Given a topological operad, weconstruct a comonad in pointed topological spaces endowed with the wedgeproduct. We then prove an approximation theorem that shows that the comonadassociated to the little $n$-cubes operad is weakly equivalent to the comonad$\Sigma^n \Omega^n$ arising from the suspension-loop space adjunction. Finally,our recognition theorem states that every little $n$-cubes coalgebra ishomotopy equivalent to an $n$-fold suspension. These results are theEckmann--Hilton dual of May's foundational results on iterated loop spaces.

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