#### Volume 20, issue 4 (2020)

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### Lukas Müller and Lukas Woike

Algebraic & Geometric Topology 20 (2020) 2029–2070
##### Abstract

Hurwitz spaces are homotopy quotients of the braid group action on the moduli space of principal bundles over a punctured plane. By considering a certain model for this homotopy quotient we build an aspherical topological operad that we call the little bundles operad. As our main result, we describe this operad as a groupoid-valued operad in terms of generators and relations and prove that the categorical little bundles algebras are precisely braided $G$–crossed categories in the sense of Turaev. Moreover, we prove that the evaluation on the circle of a homotopical two-dimensional equivariant topological field theory yields a little bundles algebra up to coherent homotopy.

##### Keywords
operad, topological field theory, braid group, monoidal category, braided monoidal category
##### Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 18D10, 57R56