Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909447975075840 |
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| author | Stewart, Natalie |
| author_facet | Stewart, Natalie |
| contents | We advance the foundational study of be Nardin-Shah's $\infty$-category of $G$-operads and their associated $\infty$-categories of algebras. In particular, we construct the underlying $G$-symmetric sequence of a (one color) $G$-operad, yielding a monadic functor; we use this to lift Bonventre's genuine operadic nerve to a conservative functor of $\infty$-categories, restricting to an equivalence between categories of discrete $G$-operads. Using this, we extend Blumberg-Hill's program concerning $\mathcal{N}_\infty$-operads to arbitrary sub-operads of the terminal $G$-operad, which we show are equivalent to weak indexing systems.
We then go on to define and characterize a homotopy-commutative and closed Boardman-Vogt tensor product on $\mathrm{Op}_G$; in particular, this specializes to a $G$-symmetric monoidal $\infty$-category of $\mathcal{O}$-algebras in a $G$-symmetric monoidal $\infty$-category whose $\mathcal{P}$-algebras are objects with interchanging $\mathcal{O}$-algebra and $\mathcal{P}$-algebra structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02129 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products Stewart, Natalie Category Theory Algebraic Topology We advance the foundational study of be Nardin-Shah's $\infty$-category of $G$-operads and their associated $\infty$-categories of algebras. In particular, we construct the underlying $G$-symmetric sequence of a (one color) $G$-operad, yielding a monadic functor; we use this to lift Bonventre's genuine operadic nerve to a conservative functor of $\infty$-categories, restricting to an equivalence between categories of discrete $G$-operads. Using this, we extend Blumberg-Hill's program concerning $\mathcal{N}_\infty$-operads to arbitrary sub-operads of the terminal $G$-operad, which we show are equivalent to weak indexing systems. We then go on to define and characterize a homotopy-commutative and closed Boardman-Vogt tensor product on $\mathrm{Op}_G$; in particular, this specializes to a $G$-symmetric monoidal $\infty$-category of $\mathcal{O}$-algebras in a $G$-symmetric monoidal $\infty$-category whose $\mathcal{P}$-algebras are objects with interchanging $\mathcal{O}$-algebra and $\mathcal{P}$-algebra structures. |
| title | Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2501.02129 |