vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Given a 2-ring , a line 2-bundle or 2-line bundle is a 2-module bundle whose typical fiber is a 2-line over .
Let be a commutative ring, or more generally an E-∞ ring. By the discussion at 2-vector space consider the 2-category
equivalent to that whose objects are associative algebras (or generally algebras) over , (being placeholders for the 2-vector space which is the category of modules over ) whose 1-morphisms are bimodules between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose 2-morphisms are homomorphisms between those.
Under Isbell duality and by the discussion at Modules – as generalized vector bundles we may think of this 2-category as being that of (generalized) 2-vector bundles over a space called .
The 2-category is canonically a monoidal 2-category. An object in is a line if it is an invertible object with respect to this tensor product, hence if it is an Azumaya algebra. In terms of the above this means that it represents a 2-vector bundle over which is a line 2-bundle.
The full inclusion
of the maximal 2-groupoid on the line 2-bundles over is a braided 3-group, the Picard 3-group of . See Relation to Brauer group below for more.
The braided 3-group of line 2-bundles over has as homotopy groups
the Brauer group of ;
the Picard group of , hence the group of ordinary line bundles over ;
the group of units of .
See at super line 2-bundle.
moduli spaces of line n-bundles with connection on -dimensional
Line 2-bundles in supergeometry as a model for the B-field and orientifolds are discussed (even if not quite explicitly in the language of higher bundles) in
based on
and
and developing constructions in
More on super line 2-bundles is secretly in
The above higher supergeometric story is made explicit in
Last revised on September 3, 2014 at 12:31:59. See the history of this page for a list of all contributions to it.