标题： 等变高阶代数中的Eckmann-Hilton论证 显示英文标题

标题： Eckmann-Hilton arguments in equivariant higher algebra

摘要： 设$\mathcal{O}^{\otimes}$和$\mathcal{P}^{\otimes}$为满足对于所有$S$的条件，$\mathcal{O}(S) = \emptyset$当且仅当$\mathcal{P}(S) = \emptyset$的$k$-和$\ell$-连通的单位$G$-操作符。 我们证明Boardman-Vogt张量积$\mathcal{O}^{\otimes} \otimes \mathcal{P}^{\otimes}$是$(k + \ell + 2)$-连通的；等价地，任何$(k + \ell + 3)$-范畴中的$\mathcal{O} \otimes \mathcal{P}$-独异点唯一地提升为不完整的半Mackey函子。 作为结果，我们证明了单位$G$-操作数上的 smash 局部化精确对应于单位$\mathcal{N}_\infty$-操作数，因此通过作者之前的的工作对应于单位弱索引系统的（有限）偏序集。 在过程中，我们将单位元$G$-操作数$\mathcal{O}^{\otimes}$的$\ell$-连通性等价地表征为$\ell$-连通性$\mathcal{O}$-可接受的 Wirthmüller 映射的$\mathcal{O}$-单子空间。 在离散情况下，在没有任何连通性假设下，$\mathcal{O} \otimes \mathcal{P}$-独异点唯一地提升为不完全半-Mackey函子，恢复了针对“$C_p$-单位的幂等代数”的Eckmann-Hilton论证。 在无限张量幂的极限情况下，我们从等变无限环空间理论中移除环路，构建了任意传递系统上不完全稳定的$G$-谱的代数逼近。 显示英文摘要

摘要： 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 Wirthm\"uller 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.