A recognition principle for iterated suspensions as coalgebras over the little cubes operad
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917296078848000 |
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| author | Flynn-Connolly, Oisín Moreno-Fernández, José M. Wierstra, Felix |
| author_facet | Flynn-Connolly, Oisín Moreno-Fernández, José M. Wierstra, Felix |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_00839 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A recognition principle for iterated suspensions as coalgebras over the little cubes operad Flynn-Connolly, Oisín Moreno-Fernández, José M. Wierstra, Felix Algebraic Topology 18M75, 55P40, 55P48 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. |
| title | A recognition principle for iterated suspensions as coalgebras over the little cubes operad |
| topic | Algebraic Topology 18M75, 55P40, 55P48 |
| url | https://arxiv.org/abs/2210.00839 |