The Space of Traces in Symmetric Monoidal Infinity Categories

IF 0.6 4区 数学 Q3 MATHEMATICS
Jan Steinebrunner
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引用次数: 1

Abstract

We define a tracelike transformation to be a natural family of conjugation invariant maps $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(\unicode{x1D7D9},\unicode{x1D7D9})$ for all dualizable objects x in any symmetric monoidal $\infty$ -category $\mathtt{C}$ . This generalizes the trace from linear algebra that assigns a scalar $\operatorname{Tr}(\,f\,) \in k$ to any endomorphism f : V → V of a finite-dimensional k-vector space. Our main theorem computes the moduli space of tracelike transformations using the one-dimensional cobordism hypothesis with singularities. As a consequence, we show that the trace $\operatorname{Tr}$ can be uniquely extended to a tracelike transformation up to a contractible space of choices. This allows us to give several model-independent characterizations of the $\infty$ -categorical trace.By restricting the aforementioned notion of tracelike transformations from endomorphisms to automorphisms one can in particular recover a theorem of Toën and Vezzosi. Other examples of tracelike transformations are for instance given by $f \mapsto \operatorname{Tr}(\,f^{\,n})$ . Unlike for $\operatorname{Tr}$ , the relevant connected component of the moduli space is not contractible, but rather equivalent to $B\mathbb{Z}/n\mathbb{Z}$ or BS 1 for n = 0. As a result, we obtain a $\mathbb{Z}/n\mathbb{Z}$ -action on $\operatorname{Tr}(\,f^{\,n})$ as well as a circle action on $\operatorname{Tr}(\operatorname{id}_x)$ .
对称单无穷范畴中的迹空间
我们将仿迹变换定义为任意对称单轴$\infty$ -范畴$\mathtt{C}$中所有可对偶对象x的共轭不变映射的自然族$T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(\unicode{x1D7D9},\unicode{x1D7D9})$。这推广了将标量$\operatorname{Tr}(\,f\,) \in k$赋给有限维k向量空间的任意自同态f: V→V的线性代数迹。我们的主要定理是利用带奇异点的一维协方差假设来计算类迹变换的模空间。因此,我们证明了迹$\operatorname{Tr}$可以唯一地扩展到一个类迹变换,直至一个可收缩的选择空间。这允许我们给出$\infty$ -分类跟踪的几个与模型无关的特征。通过将前面提到的从自同态到自同态的类迹变换的概念限制在一定范围内,我们可以特别地恢复Toën和Vezzosi的定理。其他类似跟踪的转换的例子例如由$f \mapsto \operatorname{Tr}(\,f^{\,n})$给出。与$\operatorname{Tr}$不同,模空间的相关连通分量是不可收缩的,而是等价于$B\mathbb{Z}/n\mathbb{Z}$或n = 0时的BS1。结果,我们在$\operatorname{Tr}(\,f^{\,n})$上得到一个$\mathbb{Z}/n\mathbb{Z}$ -作用,在$\operatorname{Tr}(\operatorname{id}_x)$上得到一个圆作用。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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