The Little Bundles Operad

TL;DR

This paper introduces the little bundles operad, an aspherical topological operad derived from Hurwitz spaces, and establishes its connection to Turaev's crossed categories and topological field theories.

Contribution

It constructs the little bundles operad as a groupoid-valued operad and characterizes its algebras as Turaev's crossed categories, linking operad theory and topological quantum field theories.

Findings

The little bundles operad is described by generators and relations.

Categorical little bundles algebras are equivalent to Turaev's crossed categories.

Evaluation of 2D equivariant topological field theories yields little bundles algebras.

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 Turaev's crossed categories. Moreover, we prove that the evaluation on the circle of a homotopical two-dimensional equivariant topolological field theory yields a little bundles algebra up to coherent homotopy.