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Two-vector bundles Christian Ausoni

Summary: Two-vector bundles
Christian Ausoni
(joint work with Bjørn Ian Dundas and John Rognes)
Two-vector bundles, as defined by Baas, Dundas and Rognes [6], are a 2-categorical
analogue of ordinary complex vector bundles. A two-vector bundle of rank n
over a space X can be thought of as a locally trivial bundle of categories with
fibre Vn
, where V is the bimonoidal category of finite dimensional complex vector
spaces and isomorphisms. It can be defined by means of an open cover of X
by charts with specified trivialisations, gluing data which represents a weakly
invertible matrix of ordinary vector bundles on the intersection of two charts, and
coherence isomorphisms on the intersection of three charts [6, §2]. Equivalence
classes of two-vector bundles of rank n over a finite CW-complex X are in bijective
correspondence with homotopy classes of maps from X to |B GLn(V)| [5]. By
group-completing with respect to the direct sum of matrices we obtain the space
K(V) = B
|B GLn(V)|
that represents virtual two-vector bundles. Gerbes with band U(1) coincide with
two-vector bundles of rank 1.


Source: Ausoni, Christian - Institut für Mathematische Statistik, Westfälische Wilhelms Universität Münster


Collections: Mathematics