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A parallel section functor for 2-vector bundles

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Christoph Schweigert and Lukas Woike

We associate to a 2-vector bundle over an essentially finite groupoid a
2-vector space of parallel sections, or, in representation theoretic
terms, of higher invariants, which can be described as homotopy fixed
points. Our main result is the extension of this assignment to a symmetric
monoidal 2-functor $Par : 2VecBunGrpd \to 2Vect$. It is defined on the
symmetric monoidal bicategory $2VecBunGrpd$ whose morphisms arise
from spans of groupoids in such a way that the functor
$Par$ provides pull-push maps between 2-vector spaces of
parallel sections of 2-vector bundles. The direct motivation for our
construction comes from the orbifoldization of extended equivariant
topological field theories.

Keywords:
parallel section, homotopy fixed points, higher representation, higher
vector bundle, groupoid, topological field theory

2010 MSC:
18D05, 18D10

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 23, pp 644-690.

Published 2018-06-28.

http://www.tac.mta.ca/tac/volumes/33/23/33-23.pdf

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