扩展场论是局部的且具有分类空间
Extended field theories are local and have classifying spaces
AI总结：
本文证明所有扩展函子场论均为局部的，通过构造带几何数据的配边(∞,d)-范畴并证明其余下降性质，进而建立函子场论协和类的分类空间，解决了Stolz-Teichner猜想，并据此发展幂运算的几何理论。
摘要：
我们证明所有扩展函子场论，包括拓扑的和非拓扑的，都是局部的。我们定义了带有几何数据（如黎曼度量或几何弦结构）的配边光滑(∞,d)-范畴，并证明它关于目标S满足余下降，这蕴含了局部性。我们应用这一结果构造了带几何数据的函子场论协和类的分类空间，解决了Stolz和Teichner关于此类空间存在性的猜想。我们利用分类空间构造，在Barthel、Berwick-Evans和Stapleton近期工作的基础上，发展了幂运算的几何理论。
英文摘要：
We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and prove that it satisfies codescent with respect to the target S, which implies the locality property. We apply this result to construct a classifying space for concordance classes of functorial field theories with geometric data, solving a conjecture of Stolz and Teichner about the existence of such a space. We use our classifying space construction to develop a geometric theory of power operations, following the recent work of Barthel, Berwick-Evans, and Stapleton.
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