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

Oisín Flynn-Connolly, José M. Moreno-Fernández, Felix Wierstra

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 $Σ^n Ω^n$ 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.

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

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 -cubes operad is weakly equivalent to the comonad arising from the suspension-loop space adjunction. Finally, our recognition theorem states that every little -cubes coalgebra is homotopy equivalent to an -fold suspension. These results are the Eckmann--Hilton dual of May's foundational results on iterated loop spaces.
Paper Structure (17 sections, 38 theorems, 204 equations, 3 figures)

This paper contains 17 sections, 38 theorems, 204 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Coalgebras over topological operads
  3. Construction of topological comonads
  4. Alternative definitions of a coalgebra over an operad
  5. The comonad associated to the little n-cubes operad
  6. Reduced topological operads and weak equivalences
  7. Iterated suspensions are coalgebras over the little cubes operad
  8. The Approximation Theorem
  9. The cubical support of a map and its center
  10. Proof of Theorem \ref{['teo: Approx Theorem']}
  11. The Recognition Principle for n-fold reduced suspensions
  12. The change of coalgebra structure induced by a comonad morphism
  13. The SnOn-coalgebras are n-fold reduced suspensions
  14. A point-set description of the recognition principle
  15. The map a is a morphism of comonads
  16. ...and 2 more sections

Key Result

Theorem A

The $n$-fold reduced suspension of a pointed space $X$ is a $\mathcal{C}_n$-coalgebra. More precisely, there is a natural and explicit operad map where $\operatorname{CoEnd}_{\Sigma^nX}$ is the coendomorphism operad of $\Sigma^nX$. The map $\nabla$ encodes the homotopy coassociativity and homotopy cocommutativity of the classical pinch map $\Sigma^nX \to \Sigma^nX \vee \Sigma^nX$. In particular,

Figures (3)

  • Figure 1: A little 2-cube, a singleton, and two little 2-cubes squeezed at different dimensions in $\overline{\mathcal{C}_2(1)}$.
  • Figure 2: Left: The line $\ell_c$. Right: The lines $r$ and $s$, and the midpoint $\operatorname{Mid}(c)$.
  • Figure 3: Left: The region where $G =\mathop{\mathrm{id}}\nolimits$ in red. Right: The image $G(c)$ using the proportion factor.

Theorems & Definitions (86)

  • Theorem A
  • Theorem B
  • Corollary
  • Theorem C
  • Remark 2.1
  • Definition 2.3
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 76 more