 
Summary: Twovector bundles
Christian Ausoni
(joint work with Bjørn Ian Dundas and John Rognes)
Twovector bundles, as defined by Baas, Dundas and Rognes [6], are a 2categorical
analogue of ordinary complex vector bundles. A twovector 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 twovector bundles of rank n over a finite CWcomplex X are in bijective
correspondence with homotopy classes of maps from X to B GLn(V) [5]. By
groupcompleting with respect to the direct sum of matrices we obtain the space
K(V) = B
n
B GLn(V)
that represents virtual twovector bundles. Gerbes with band U(1) coincide with
twovector bundles of rank 1.
