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The Space of Traces in Symmetric Monoidal Infinity Categories
Quarterly Journal of Mathematics ( IF 0.6 ) Pub Date : 2021-02-23 , DOI: 10.1093/qmath/haab013
Jan Steinebrunner 1
Affiliation  

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 BS1 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 $\operatorname{Tr}(\operatorname{id}_x)$.

中文翻译:

对称 Monoidal Infinity 范畴中的迹空间

我们将类迹变换定义为共轭不变映射的自然族 $T_{x,\mathtt{C}}:\hom_\mathtt{C}(x, x) \to \hom_\mathtt{C}(\unicode {x1D7D9},\unicode{x1D7D9})$ 用于任何对称幺半群 $\infty$-category $\mathtt{C}$ 中的所有可对偶对象 x。这概括了线性代数的迹,它将标量 $\operatorname{Tr}(\,f\,) \in k$ 分配给有限维 k 向量空间的任何自同态 f : V → V。我们的主要定理使用具有奇点的一维协调假设来计算迹线变换的模空间。因此,我们证明了迹线 $\operatorname{Tr}$ 可以唯一地扩展到类似迹线的转换,直至可收缩的选择空间。这使我们能够给出 $\infty$-categorical trace 的几个与模型无关的特征。通过限制上述从自同构到自同构的迹状变换的概念,我们可以特别恢复 Toën 和 Vezzosi 的定理。其他类似轨迹变换的例子由 $f \mapsto \operatorname{Tr}(\,f^{\,n})$ 给出。与 $\operatorname{Tr}$ 不同,模空间的相关连通分量不可收缩,而是等价于 $B\mathbb{Z}/n\mathbb{Z}$ 或 n = 0 时的 BS1。作为结果,我们获得了对 $\operatorname{Tr}(\,f^{\,n})$ 的 $\mathbb{Z}/n\mathbb{Z}$-action 以及对 $\operatorname 的循环动作{Tr}(\operatorname{id}_x)$。其他类似轨迹变换的例子由 $f \mapsto \operatorname{Tr}(\,f^{\,n})$ 给出。与 $\operatorname{Tr}$ 不同,模空间的相关连通分量不可收缩,而是等价于 $B\mathbb{Z}/n\mathbb{Z}$ 或 n = 0 时的 BS1。作为结果,我们获得了对 $\operatorname{Tr}(\,f^{\,n})$ 的 $\mathbb{Z}/n\mathbb{Z}$-action 以及对 $\operatorname 的循环动作{Tr}(\operatorname{id}_x)$。其他类似轨迹变换的例子由 $f \mapsto \operatorname{Tr}(\,f^{\,n})$ 给出。与 $\operatorname{Tr}$ 不同,模空间的相关连通分量不可收缩,而是等价于 $B\mathbb{Z}/n\mathbb{Z}$ 或 n = 0 时的 BS1。作为结果,我们获得了对 $\operatorname{Tr}(\,f^{\,n})$ 的 $\mathbb{Z}/n\mathbb{Z}$-action 以及对 $\operatorname 的循环动作{Tr}(\operatorname{id}_x)$。
更新日期:2021-02-23
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