Equivariant Homotopy Theory via Simplicial Coalgebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912312417320960 |
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| author | Alberga, Sofía Martínez Rivera, Manuel |
| author_facet | Alberga, Sofía Martínez Rivera, Manuel |
| contents | Given a commutative ring $R$, a $π_1$-$R$-equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers. When $R$ is an algebraically closed field, Raptis and Rivera described a full and faithful model for the homotopy theory of spaces up to $π_1$-$R$-equivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by a localized version of the Cobar functor. In this article, we prove a $G$-equivariant analog of this statement using a generalization of a celebrated theorem of Elmendorf. We also prove a more general result about modeling $G$-simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04688 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Homotopy Theory via Simplicial Coalgebras Alberga, Sofía Martínez Rivera, Manuel Algebraic Topology Given a commutative ring $R$, a $π_1$-$R$-equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers. When $R$ is an algebraically closed field, Raptis and Rivera described a full and faithful model for the homotopy theory of spaces up to $π_1$-$R$-equivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by a localized version of the Cobar functor. In this article, we prove a $G$-equivariant analog of this statement using a generalization of a celebrated theorem of Elmendorf. We also prove a more general result about modeling $G$-simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras. |
| title | Equivariant Homotopy Theory via Simplicial Coalgebras |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2410.04688 |