Smooth 2-群的中心扩张与一维弦2-群
Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group
摘要：
我们提供了一个模型的弦群作为一个中心的扩展有限维2群的双范畴李群胚，左主双丛，双丛映射。这个双范畴是光滑栈双范畴的几何化身，并推广了李群胚、光滑函子和光滑自然变换的更朴素的2-范畴。特别是这个概念的顺利2-群包含的概念李2-群介绍了贝兹劳达。更确切地说，我们分类的一个大家庭，这些中心扩张的拓扑群上同调G。Segal，而我们的String 2-群是这种扩张的一个特例。有一个神经结构，可以应用于这些2-群，以获得一个单纯流形，允许与A的模型进行比较。亨里克斯几何实现是一个$A_\infty$ -空间，在我们的模型中，具有正确的同伦类型String（n）。与所有以前的模型，我们的建设完全发生在有限维流形和李群胚的框架内。此外，在这种情况下，我们的模型的特点是一个强大的唯一性结果。它是Spin（n）的唯一中心扩展。
英文摘要：
We provide a model of the String group as a central extension of finite-dimensional 2-groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naïve 2-category of Lie groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth 2-group subsumes the notion of Lie 2-group introduced by Baez-Lauda. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by G. Segal, and our String 2-group is a special case of such extensions. There is a nerve construction which can be applied to these 2-groups to obtain a simplicial manifold, allowing comparison with with the model of A. Henriques. The geometric realization is an $A_\infty$-space, and in the case of our model, has the correct homotopy type of String(n). Unlike all previous models our construction takes place entirely within the framework of finite dimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a unique central extension of Spin(n).
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