Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group
Christopher Schommer-Pries

TL;DR
This paper constructs a finite-dimensional model of the String 2-group as a central extension of Lie 2-groups, providing a geometric and algebraic framework that generalizes previous models and ensures the correct homotopy type.
Contribution
It introduces a finite-dimensional, geometric model of the String 2-group as a central extension of Lie 2-groups within Lie groupoids, unifying and extending prior approaches.
Findings
Provides a classification of central extensions via topological group cohomology.
Constructs a nerve that yields a simplicial manifold comparable to existing models.
Establishes a unique, finite-dimensional model with the correct homotopy type of String(n).
Abstract
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 na\"ive 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 -space, and in the case of…
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