Point-set models for homotopy coherent coalgebras
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913046798008320 |
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| author | Petersen, Dan Lucio, Victor Roca i Yalin, Sinan |
| author_facet | Petersen, Dan Lucio, Victor Roca i Yalin, Sinan |
| contents | We show a first rectification result for homotopy chain coalgebras over a field. On the one hand, we consider the $\infty$-category obtained by localizing differential graded coalgebras over an operad with respect to quasi-isomorphisms; on the other, we give a general definition of an $\infty$-category of coalgebras over an enriched $\infty$-operad. We show by induction over cell attachments that these two $\infty$-categories are in fact equivalent when the operad is cofibrant. This yields explicit point-set models for $\mathbb{E}_n$-coalgebras and $E_\infty$-coalgebras in the derived $\infty$-category of chain complexes over a field, and an explicit point-set model for the cellular chains functor with its $E_\infty$-coalgebra structure. After Bachmann--Burklund, this gives a point-set algebraic model for nilpotent $p$-adic homotopy types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03101 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Point-set models for homotopy coherent coalgebras Petersen, Dan Lucio, Victor Roca i Yalin, Sinan Algebraic Topology Category Theory Quantum Algebra Rings and Algebras 18N70, 18M60, 18N40, 18N60, 55U15, 55U35, 16T15 We show a first rectification result for homotopy chain coalgebras over a field. On the one hand, we consider the $\infty$-category obtained by localizing differential graded coalgebras over an operad with respect to quasi-isomorphisms; on the other, we give a general definition of an $\infty$-category of coalgebras over an enriched $\infty$-operad. We show by induction over cell attachments that these two $\infty$-categories are in fact equivalent when the operad is cofibrant. This yields explicit point-set models for $\mathbb{E}_n$-coalgebras and $E_\infty$-coalgebras in the derived $\infty$-category of chain complexes over a field, and an explicit point-set model for the cellular chains functor with its $E_\infty$-coalgebra structure. After Bachmann--Burklund, this gives a point-set algebraic model for nilpotent $p$-adic homotopy types. |
| title | Point-set models for homotopy coherent coalgebras |
| topic | Algebraic Topology Category Theory Quantum Algebra Rings and Algebras 18N70, 18M60, 18N40, 18N60, 55U15, 55U35, 16T15 |
| url | https://arxiv.org/abs/2601.03101 |