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
–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.
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