On tensor products with equivariant commutative operads
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916883644547072 |
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| author | Stewart, Natalie |
| author_facet | Stewart, Natalie |
| contents | We affirm and generalize a conjecture of Blumberg and Hill: unital weak $\mathcal{N}_\infty$-operads are closed under $\infty$-categorical Boardman-Vogt tensor products and the resulting tensor products correspond with joins of weak indexing systems; in particular, we acquire a natural $G$-symmetric monoidal equivalence
\[
\underline{\mathrm{CAlg}}^{\otimes}_{I} \underline{\mathrm{CAlg}}^{\otimes}_{J} \mathcal{C} \simeq \underline{\mathrm{CAlg}}^{\otimes}_{I \vee J} \mathcal{C}.
\]
We accomplish this by showing that $\mathcal{N}_{I\infty}^{\otimes}$ is $\otimes$-idempotent and $\mathcal{O}^{\otimes}$ is local for the corresponding smashing localization if and only if $\mathcal{O}$-monoid $G$-spaces satisfy $I$-indexed Wirthmüller isomorphisms.
Ultimately, we accomplish this by advancing the equivariant higher algebra of cartesian and cocartesian $I$-symmetric monoidal $\infty$-categories. Additionally, we acquire a number of structural results concerning $G$-operads, including a canonical lift of $\otimes$ to a presentably symmetric monoidal structure and a general disintegration and assembly procedure for computing tensor products of non-reduced unital $G$-operads. All such results are proved in the generality of atomic orbital $\infty$-categories.
We also achieve the expected corollaries for (iterated) Real topological Hochschild and cyclic homology and construct a natural $I$-symmetric monoidal structure on right modules over an $\mathcal{N}_{I\infty}$-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On tensor products with equivariant commutative operads Stewart, Natalie Algebraic Topology Category Theory We affirm and generalize a conjecture of Blumberg and Hill: unital weak $\mathcal{N}_\infty$-operads are closed under $\infty$-categorical Boardman-Vogt tensor products and the resulting tensor products correspond with joins of weak indexing systems; in particular, we acquire a natural $G$-symmetric monoidal equivalence \[ \underline{\mathrm{CAlg}}^{\otimes}_{I} \underline{\mathrm{CAlg}}^{\otimes}_{J} \mathcal{C} \simeq \underline{\mathrm{CAlg}}^{\otimes}_{I \vee J} \mathcal{C}. \] We accomplish this by showing that $\mathcal{N}_{I\infty}^{\otimes}$ is $\otimes$-idempotent and $\mathcal{O}^{\otimes}$ is local for the corresponding smashing localization if and only if $\mathcal{O}$-monoid $G$-spaces satisfy $I$-indexed Wirthmüller isomorphisms. Ultimately, we accomplish this by advancing the equivariant higher algebra of cartesian and cocartesian $I$-symmetric monoidal $\infty$-categories. Additionally, we acquire a number of structural results concerning $G$-operads, including a canonical lift of $\otimes$ to a presentably symmetric monoidal structure and a general disintegration and assembly procedure for computing tensor products of non-reduced unital $G$-operads. All such results are proved in the generality of atomic orbital $\infty$-categories. We also achieve the expected corollaries for (iterated) Real topological Hochschild and cyclic homology and construct a natural $I$-symmetric monoidal structure on right modules over an $\mathcal{N}_{I\infty}$-algebra. |
| title | On tensor products with equivariant commutative operads |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2504.02143 |