Algebraic & Geometric Topology

The little bundles operad

Lukas Müller and Lukas Woike

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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

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

Received: 12 March 2019
Revised: 21 September 2019
Accepted: 1 October 2019
First available in Project Euclid: 1 August 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48]
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 57R56: Topological quantum field theories

operad topological field theory braid group monoidal category braided monoidal category


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

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