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

TL;DR

This paper establishes a recognition principle linking iterated suspensions to coalgebras over the little disks operad, providing a dual perspective to known loop space results in algebraic topology.

Contribution

It introduces a new recognition theorem for iterated suspensions as coalgebras over the little cubes operad, extending the algebraic topology framework.

Findings

Comonad associated with little n-cubes operad is weakly equivalent to suspension-loop comonad.

Every little n-cubes coalgebra is homotopy equivalent to an n-fold suspension.

Provides a dual perspective to May's results on iterated loop spaces.

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 $\Sigma^n \Omega^n$ 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.