## Algebraic & Geometric Topology

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 20, Number 4 (2020), 2029-2070.

Dates
Revised: 21 September 2019
Accepted: 1 October 2019
First available in Project Euclid: 1 August 2020

https://projecteuclid.org/euclid.agt/1596247245

Digital Object Identifier
doi:10.2140/agt.2020.20.2029

Mathematical Reviews number (MathSciNet)
MR4127089

Zentralblatt MATH identifier
07226710

#### Citation

Müller, Lukas; Woike, Lukas. The little bundles operad. Algebr. Geom. Topol. 20 (2020), no. 4, 2029--2070. doi:10.2140/agt.2020.20.2029. https://projecteuclid.org/euclid.agt/1596247245

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