Principal $\infty$-Bundles and Smooth String Group Models

TL;DR

This paper develops a homotopy-theoretic framework for defining and constructing smooth models of the string group within an $ abla$-category of smooth spaces, introducing new models and characterizations of principal $ abla$-bundles.

Contribution

It provides a general homotopy-theoretic definition of smooth string group models and constructs new models within the $ abla$-category of smooth spaces, extending previous work.

Findings

New smooth models for the string group are constructed.

Characterizations of principal $ abla$-bundles and group extensions are provided.

Obstructions to equivariant structures on gerbes lead to these new models.

Abstract

We provide a general, homotopy-theoretic definition of string group models within an $\infty$-category of smooth spaces, and we present new smooth models for the string group. Here, a smooth space is a presheaf of $\infty$-groupoids on the category of cartesian spaces. The key to our definition and construction of smooth string group models is a version of the singular complex functor, which assigns to a smooth space an underlying ordinary space. We provide new characterisations of principal $\infty$-bundles and group extensions in $\infty$-topoi, building on work of Nikolaus, Schreiber, and Stevenson. These insights allow us to transfer the definition of string group extensions from the $\infty$-category of spaces to the $\infty$-category of smooth spaces. Finally, we consider smooth higher-categorical group extensions that arise as obstructions to the existence of equivariant structures on gerbes. We show that these extensions give rise to new smooth models for the string group, as recently conjectured in joint work with M\"uller and Szabo.