under construction
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.
We are looking for a generalization of the notion of vector bundle in higher category theory.
Let $X$ be a topological space and $\{U_\alpha \to X\}_{\alpha \in A}$ an open cover, where the index set $A$ is assumed to be a poset.
Defition. A charted 2-vector bundle (i.e. a cocycle for a BDR 2-vector bundle) of rank $n$ is
for $\alpha \lt \beta \in A$ on $U_\alpha \cap U_\beta =: U_{\alpha\beta}$ a matrix $(E^{\alpha\beta}_{i j})_{i,j = 1}^{n}$ of vector bundles $E^{\alpha \beta}_{i j} \to U_{\alpha \beta}$ such that the determinant of the underlying matrix of dimensions is $det(dim(E^{\alpha \beta}_{i j})) = \pm 1$.
on triple overlaps $U_{\alpha \beta \gamma}$ for $\alpha \lt \beta \lt \gamma \in A$ we have isomorphisms
such that the $\phi$ 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 $2Vect(X)$ for the equivalence classes of charted 2-vector bundles under these morphisms.
Remark If we restrict attention to $n = 1$ then this gives the same as $U(1)$-gerbes/bundle gerbes.
Theorem (Baas-Dundas-Rognes)
There exists a classifying space $\mathcal{K}(V)$ such that for $X$ a finite CW-complex there is an isomorphism
between homotopy classes of continuous maps $X \to \mathcal{K}(V)$ 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 $Gr(-)$ indicates the Grothendieck group completion using the monoid structure arising from the direct sum of 2-vector bundles.
Proof In BDR, Segal Birthday Proceedings
Note $2Vect_n(X) = [X, |B Gl_n (V)| ]$.
BDR called $\mathcal{K}(V)$ the 2-K-theory of the bimonoidal category of Kapranov-Voevodsky 2-vector spaces.
Theorem (Baas-Dundas-Rognes-Richter)
$\mathcal{K}(V) \simeq K(ku)$
Here:
$ku$ is the connected version of the spectrum of complex topological K-theory;
$K(-)$ denotes forming the algebraic K-theory spectrum of ring spectra.
So by the general formula for algebraic K-theory for ring spectra, this is
Some flavor of $\mathcal{K}(V)$.
$V$ is the (a skeleton of the core of) of category of complex vector spaces.
objects are natural numbers, $n$ corresponding to $\mathbb{C}^n$;
morphisms are $Hom(k,k) = GL(k)$ and there are no morphisms between different $k,l$.
This category $V$ 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 $Mat_n(V)$ for the $n \times n$ 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 $Gl_n(V)$ for the subcollection of those matrices for which the determinant of their matrix of dimensions is $\pm 1$.
Now define
Notice that this is a direct generalization of the corresponding formula for the algebraic K-theory of a ring $R$,
Ausoni and Rognes compute the homology groups (for a certain sense of homology) of $K(ku)$.
take rational homotopy
$H^*(-, \mathbb{Q})$
for $p$ a prime, multiplying by $p$ gives an isomorphism on this.
p = $\nu_0$
Let $KU^*(-)$ be complex oriented topological K-theory, then
for $|u| = 2$ (the Bott class) we have that multiplying by $u$ is an isomorphism and $u^{p-1} = \nu_1$
The $\nu_i$ come from the Brown-Peterson spectrum $B P$ and $\pi_* BP = \mathbb{Z}_{(p)}[\nu_1, \nu_2, \cdots]$
motto: the higher the $\nu_i$ the more you detect.
Christian Ausoni figured out something that implies that
“$K(ku)$ detects as much in the stable homotopy category as any other form of elliptic cohomology.”
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 $\Sigma B U(1) \to B BU(1) \to B B U_{\otimes}$ we can take a $U(1)$-gerbe classified by maps into $B^2 U(1)$ 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 $\mu : K(\mathbb{Z},3) \to K(ku)$ we have
Fact: $\mu$ gives a generator in $\pi_3 K(\mathbb{Z},3) = H^3(\mathbb{S}^3)$
Theorem (Ausoni-Dundas-Rognes)
in $\pi_3(K(ku))$
so regarded as a 2-vector bundle $\mu$ is not a generator.
ADR: $\zeta$ is “half a monopole”.
(the first summand is $\zeta$, the second $\nu$).
Thomas Krogh has an orientation theory for 2-vector bundles which says that $j(\nu)$ is not orientable.
Let $(R, \oplus, \otimes, 0,1, c_{\oplus})$ be a bimonoidal category, i.e. a categorified rig.
This can be broken down as
$(R, \oplus, 0 , c_{\oplus})$ a permutative category, a categorified abelian monoid;
$(R , \otimes, 1)$ is a monoidal category, assumed to be strict monidal in the following;
a distributivity law.
Examples
$E = Core$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 $n \in \mathb{N}$, $\oplus$ and $\otimes$ is addition and multiplication on $\mathbb{N}$, respectively. Here $c_{\oplus}$ is the evident natural isomorphism between direct sums of finite sets.
$V = Core$Vect the core of the category of finite dim vector spaces, with morphisms only between those of the same dimension.
…
Definition For $R$ a bimonoidal category, write $Mat_n(R)$ for the $n \times n$ matrices with entries morphisms in $R$. Then matrix multiplication is defined using the bimonoidal structure on $R$. This gives a weak monoid structure.
Let $Gl_n(R)$ be the category of weakly invertible such matrices. This is the full subcategory of $Mat_n(R)$. We get a diagram of pullback squares
where $Gr(-)$ denotes Grothendieck group-completion.
Definition (Baas-Dundas-Rognes, 2004)
For $R$ a bimonoidal category, the 2K-theory of $R$ is
where the $\Omega$ is forming loop space, the leftmost $B$ is forming classifying space of a category and the inner $B$ is a flabby version of classifying space of a category.
This can also be written
Here $B_q Gl_n(R)$ is a simplicial category
…
Theorem (Baas-Dundas-Rognes)
Let $R$ be a small Top-enriched bimonoidal category such that
Then $\mathcal{K}(R) \simeq K(H R)$ is the ordinary algebraic K-theory of the ring spectrum $H R$.
Notice that for $H R$ to be a spectrum we only need the additive structure $(R, \oplus, 0, c_{\oplus})$. The point is that the other monoidal structure $\otimes$ 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).
Examples
For the category $R := E = Core(FinSet)$ of finite sets as above we have that $H E$ is the sphere spectrum.
For $R := V = Core(FinVect)$ the core of complex finite dimensional vector spaces we have $H V$ is the complex K-theory spectrum.
For $V_{\mathbb{R}}$ analogously we get the real K-theory spectrum.
So by the above theorem
$\mathcal{K}(E) \simeq K(S) \simeq A(*)$
$\mathcal{K}(V) \simeq K(ku)$;
etc.
Remarks
A. Osono: The equivalence $\mathcal{K}(R) \simeq K(H R)$ of topological spaces is even an equivalence of infinity loop space?s;
Application of that:
a) for $E$ a ring spectrum: find a model
and use the equivalence $\mathbb{K}(R) \simeq K(H R E)$ to understand arithmetic properties of $E$.
b) Often one knows $K(H(R))$ via calculations. here $\mathcal{K}(R)$ might help to get some deeper understanding.
c) Theorem (Birgit Richter): for $R$ a bimonoidal category with anti-involution, then you get an involution of $\mathcal{K}(R)$.
The original articles on BDR 2-vector bundles are
Their classifying spaces are discussed in
Nils Baas, Ian Dundas, Birgit Richter, John Rognes, Ring completion of rig categories (arXiv:0706.0531)
(a previous version of this carried the title Two-vector bundles define a form of elliptic cohomology)
Nils Baas, M. Böckstedt, Tore Kro, Two-Categorical Bundles and Their Classifying Spaces (2008) (arXiv:math/0612549), .
Orientation of BDR 2-vector bundles is discussed in