关于切片的范畴
On categories of slices
摘要：
本文给出了任意群$G$的$n$ -片范畴在Hill-Hopkins-Ravenel意义下的代数描述.具体地说，给定有限群$G$和整数$n$，我们构造了一个显式的$G$ -谱$W$（i）$G$ -谱$X$的$n$ -片等价于Mackey函子$\underline{[W,X]}$作为自同态Green函子$\underline{[W,W]}$上的模的某个商的数据;（ii）$n$ -切片范畴等价于$\underline{[W,W]}$上右模的满子范畴，对它某些限制映射是内射的.我们使用这个定理来恢复已知的结果类别的切片日期，并展示我们的描述在几个新的例子中的效用。我们更进一步，并表明，绿色函子$\underline{[W,W]}$某些切片$n$ -球体有一个特殊的属性（他们是“几何分裂”），这减少了指定一个$\underline{[W,W]}$ -模块所需的数据量。这一步是纯代数的，可能是独立的利益。
英文摘要：
In this paper we give an algebraic description of the category of $n$-slices for an arbitrary group $G$, in the sense of Hill-Hopkins-Ravenel. Specifically, given a finite group $G$ and an integer $n$, we construct an explicit $G$-spectrum $W$ (called an isotropic slice $n$-sphere) with the following properties: (i) the $n$-slice of a $G$-spectrum $X$ is equivalent to the data of a certain quotient of the Mackey functor $\underline{[W,X]}$ as a module over the endomorphism Green functor $\underline{[W,W]}$; (ii) the category of $n$-slices is equivalent to the full subcategory of right modules over $\underline{[W,W]}$ for which certain restriction maps are injective. We use this theorem to recover the known results on categories of slices to date, and exhibit the utility of our description in several new examples. We go further and show that the Green functors $\underline{[W,W]}$ for certain slice $n$-spheres have a special property (they are "geometrically split") which reduces the amount of data necessary to specify a $\underline{[W,W]}$-module. This step is purely algebraic and may be of independent interest.
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。
