Eckmann-Hilton arguments in equivariant higher algebra
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913979255750656 |
|---|---|
| author | Stewart, Natalie |
| author_facet | Stewart, Natalie |
| contents | Let $\mathcal{O}^{\otimes}$ and $\mathcal{P}^{\otimes}$ be $k$- and $\ell$-connected unital $G$-operads subject to the condition for all $S$ that $\mathcal{O}(S) = \emptyset$ if and only if $\mathcal{P}(S) = \emptyset$. We show that the Boardman-Vogt tensor product $\mathcal{O}^{\otimes} \otimes \mathcal{P}^{\otimes}$ is $(k + \ell + 2)$-connected; equivalently, $\mathcal{O} \otimes \mathcal{P}$-monoids in any $(k + \ell + 3)$-category lift uniquely to incomplete semi-Mackey functors. As a consequence, we show that the smashing localizations on unital $G$-operads correspond precisely to unital $\mathcal{N}_\infty$-operads, and hence to the (finite) poset of unital weak indexing systems by previous work of the author. Along the way we characterize $\ell$-connectivity of a unital $G$-operad $\mathcal{O}^{\otimes}$ equivalently as $\ell$-connectivity of $\mathcal{O}$-admissible Wirthmüller maps of $\mathcal{O}$-monoid spaces.
In the discrete case, under no connectivity assumptions, $\mathcal{O} \otimes \mathcal{P}$-monoids lift uniquely to incomplete semi-Mackey functors, recovering an Eckmann-Hilton argument for "$C_p$-unital magmas." In the limiting case of infinite tensor powers, we take the loops out of equivariant infinite loop space theory, constructing algebraic approximations to incompletely stable $G$-spectra over arbitrary transfer systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eckmann-Hilton arguments in equivariant higher algebra Stewart, Natalie Category Theory Algebraic Topology Let $\mathcal{O}^{\otimes}$ and $\mathcal{P}^{\otimes}$ be $k$- and $\ell$-connected unital $G$-operads subject to the condition for all $S$ that $\mathcal{O}(S) = \emptyset$ if and only if $\mathcal{P}(S) = \emptyset$. We show that the Boardman-Vogt tensor product $\mathcal{O}^{\otimes} \otimes \mathcal{P}^{\otimes}$ is $(k + \ell + 2)$-connected; equivalently, $\mathcal{O} \otimes \mathcal{P}$-monoids in any $(k + \ell + 3)$-category lift uniquely to incomplete semi-Mackey functors. As a consequence, we show that the smashing localizations on unital $G$-operads correspond precisely to unital $\mathcal{N}_\infty$-operads, and hence to the (finite) poset of unital weak indexing systems by previous work of the author. Along the way we characterize $\ell$-connectivity of a unital $G$-operad $\mathcal{O}^{\otimes}$ equivalently as $\ell$-connectivity of $\mathcal{O}$-admissible Wirthmüller maps of $\mathcal{O}$-monoid spaces. In the discrete case, under no connectivity assumptions, $\mathcal{O} \otimes \mathcal{P}$-monoids lift uniquely to incomplete semi-Mackey functors, recovering an Eckmann-Hilton argument for "$C_p$-unital magmas." In the limiting case of infinite tensor powers, we take the loops out of equivariant infinite loop space theory, constructing algebraic approximations to incompletely stable $G$-spectra over arbitrary transfer systems. |
| title | Eckmann-Hilton arguments in equivariant higher algebra |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2508.05556 |