乘性等变$K$ -理论与Barratt-Priddy-Quillen定理
Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem
摘要：
我们证明了等变Barratt-Priddy-Quillen定理的乘法版本，从arXiv：1207.3459中证明的加法版本开始。证明使用了一个乘法阐述的添加剂等变无限闭路空间机器，制造正交$G$ -频谱从对称monoidal $G$ -类别。新的机器产生高度结构化的结合环和模块$G$ -频谱从适当的乘法输入。它依赖于新的操作性多范畴，这些范畴具有相当大的独立兴趣，并在一般的，不一定是等变的或拓扑的背景下定义。我们的大部分工作都集中在构建和比较它们。我们构造了一个从对称monoidal $G$ -范畴到正交$G$ -谱的多重函子.有了这个机器，我们证明了等变BPQ定理可以提升到乘法等价。这是arXiv：1110.3571中$G$ -谱类别的预层重建所需的核心。
英文摘要：
We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.
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