等变operad、对称序列与Boardman-Vogt张量积
Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products
AI总结：
本文通过构造$G$-operad的底层$G$-对称序列及保守函子，将Blumberg-Hill关于$\mathcal{N}_\infty$-operad的纲领推广到任意子operad，并定义刻画了$\mathrm{Op}_G$上同伦交换且封闭的Boardman-Vogt张量积。
摘要：
我们推进了Nardin-Shah的$G$-operad的$\infty$-范畴及其相应代数$\infty$-范畴的基础性研究。特别地，我们构造了（单色）$G$-operad的底层$G$-对称序列，得到一个单子函子；我们利用这一点将Bonventre的真诚operadic神经提升为$\infty$-范畴的保守函子，并限制为离散$G$-operad范畴之间的等价。利用这一结果，我们将Blumberg-Hill关于$\mathcal{N}_\infty$-operad的纲领推广到终$G$-operad的任意子operad，并证明它们等价于弱指标系统。随后，我们定义并刻画了$\mathrm{Op}_G$上的同伦交换且封闭的Boardman-Vogt张量积；特别地，它特殊化为$G$-对称幺半$\infty$-范畴中$\mathcal{O}$-代数的$G$-对称幺半$\infty$-范畴，其$\mathcal{P}$-代数是具有可交换的$\mathcal{O}$-代数与$\mathcal{P}$-代数结构的对象。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
