for the moment this here are nothing but rough notes taken in a talk long ago
BDR 2-vector bundles are a notion of categorified vector bundle motivated by the concept of Kapranov-Voevodsy’s 2-vector spaces.
2-Vector bundles following Baas-Dundas-Rognes
We are looking for a generalization of the notion of vector bundle in higher category theory.
Let be a topological space and an open cover, where the index set is assumed to be a poset.
Defition. A charted 2-vector bundle (i.e. a cocycle for a BDR 2-vector bundle) of rank is
for on a matrix of vector bundles such that the determinant of the underlying matrix of dimensions is .
on triple overlaps for we have isomorphisms
such that the satisfy on quadruple overlaps the evident cocycle condition (as described at gerbe and principal 2-bundle).
Next we need to define morphisms of such charted 2-vector bundles. These involve the evident refinements of covers and fiberwise transformations.
Write for the equivalence classes of charted 2-vector bundles under these morphisms.
Remark If we restrict attention to then this gives the same as -gerbes/bundle gerbes.
There exists a classifying space such that for a finite CW-complex there is an isomorphism
between homotopy classes of continuous maps and equivalence classes of the group completed stackification of 2-vector bundles,
where the colimit is over acyclic Serre fibrations (Note: these are not acyclic fibrations in the usual sense, rather their fibres have trivial integral homology) and indicates the Grothendieck group completion using the monoid structure arising from the direct sum of 2-vector bundles.
Proof In BDR, Segal Birthday Proceedings
BDR called the 2-K-theory of the bimonoidal category of Kapranov-Voevodsky 2-vector spaces.
The homotopy type of the classifying space
So by the general formula for algebraic K-theory for ring spectra, this is
Some flavor of .
This category is naturally a bimonoidal category under coproduct and tensor product of vector spaces.
K-theory is about understanding linear algebra on a ring, so we want to understand the linear algebra of this monoidal category.
We write for the matrices of linear isomorphisms between finite dimensional vector spaces.
Such matrices can be multiplied using the usual formula for matrix products on the tensor product and direct sum of vector space and linear maps.
Write for the subcollection of those matrices for which the determinant of their matrix of dimensions is .
Notice that this is a direct generalization of the corresponding formula for the algebraic K-theory of a ring ,
as a form of elliptic cohomology
Ausoni and Rognes compute the homology groups (for a certain sense of homology) of .
take rational homotopy
for a prime, multiplying by gives an isomorphism on this.
Let be complex oriented topological K-theory, then
for (the Bott class) we have that multiplying by is an isomorphism and
The come from the Brown-Peterson spectrum and
motto: the higher the the more you detect.
Christian Ausoni figured out something that implies that
“ detects as much in the stable homotopy category as any other form of elliptic cohomology.”
From gerbes to 2-vector bundles
It is hard to directly construct charted 2-vector bundles. We have more examples of gerbes. So we want to get one from the other.
Example We have
(see Eilenberg-MacLane space and classifying space)
using we can take a -gerbe classified by maps into and induce from it the associated 2-vector bundle.
the canonical map
may be thought of as classifying the gerbe called the magnetic monopole-gerbe
Postcomposing with we have
Fact: gives a generator in
so regarded as a 2-vector bundle is not a generator.
ADR: is “half a monopole”.
(the first summand is , the second ).
Thomas Krogh has an orientation theory for 2-vector bundles which says that is not orientable.
2K-theory of bimonoidal categories
Let be a bimonoidal category, i.e. a categorified rig.
This can be broken down as
a permutative category, a categorified abelian monoid;
is a monoidal category, assumed to be strict monidal in the following;
a distributivity law.
FinSet, the core of the category of finite sets and morphisms only between sets of the same cardinality.
In the skeleton, objects are natural numbers , and is addition and multiplication on , respectively. Here is the evident natural isomorphism between direct sums of finite sets.
Vect the core of the category of finite dim vector spaces, with morphisms only between those of the same dimension.
Definition For a bimonoidal category, write for the matrices with entries morphisms in . Then matrix multiplication is defined using the bimonoidal structure on . This gives a weak monoid structure.
Let be the category of weakly invertible such matrices. This is the full subcategory of . We get a diagram of pullback squares
where denotes Grothendieck group-completion.
Definition (Baas-Dundas-Rognes, 2004)
For a bimonoidal category, the 2K-theory of is
where the is forming loop space, the leftmost is forming classifying space of a category and the inner is a flabby version of classifying space of a category.
This can also be written
Here is a simplicial category
Let be a small Top-enriched bimonoidal category such that
is a groupoid;
for all we have that is faithful.
Then is the ordinary algebraic K-theory of the ring spectrum .
Notice that for to be a spectrum we only need the additive structure . The point is that the other monoidal structure indeed makes this a ring spectrum. This is a not completely trivial statement due to a bunch of people, involving Peter May and Elmendorf-Mandell (2006).
For the category of finite sets as above we have that is the sphere spectrum.
For the core of complex finite dimensional vector spaces we have is the complex K-theory spectrum.
For analogously we get the real K-theory spectrum.
So by the above theorem
A. Osono: The equivalence of topological spaces is even an equivalence of infinity loop space?s;
Application of that:
a) for a ring spectrum: find a model
and use the equivalence to understand arithmetic properties of .
b) Often one knows via calculations. here might help to get some deeper understanding.
c) Theorem (Birgit Richter): for a bimonoidal category with anti-involution, then you get an involution of .
The original articles on BDR 2-vector bundles are
- Nils Baas, Ian Dundas, John Rognes, Two-vector bundles and forms of elliptic cohomology, in: Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 18–45. Cambridge Univ. Press, Cambridge, (2004).
Their classifying spaces are discussed in
Orientation of BDR 2-vector bundles is discussed in
- Thomas Kragh, Orientations and Connective Structures on 2-vector Bundles (arXiv:0910.0131)