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The little bundles operad

Lukas Müller and Lukas Woike

Algebraic & Geometric Topology 20 (2020) 2029–2070

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.

operad, topological field theory, braid group, monoidal category, braided monoidal category
Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 18D10, 57R56
Received: 12 March 2019
Revised: 21 September 2019
Accepted: 1 October 2019
Published: 20 July 2020
Lukas Müller
Department of Mathematics
Heriot-Watt University
United Kingdom
Maxwell Institute for Mathematical Sciences
United Kingdom
Lukas Woike
Fachbereich Mathematik
Universität Hamburg