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\section*{BDR 2-vector bundle}

\begin{quote}%
under construction

for the moment this here are nothing but rough notes taken in a talk long ago


\end{quote}
\emph{BDR 2-vector bundles} are a notion of [[categorified]] [[vector bundle]] motivated by the concept of \href{http://ncatlab.org/nlab/show/2-vector+space#KV2VectorSpace}{Kapranov-Voevodsy's 2-vector spaces}.

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{2vector_bundles_following_baasdundasrognes}{2-Vector bundles following Baas-Dundas-Rognes}\dotfill \pageref*{2vector_bundles_following_baasdundasrognes} \linebreak
\noindent\hyperlink{the_homotopy_type_of_the_classifying_space}{The homotopy type of the classifying space}\dotfill \pageref*{the_homotopy_type_of_the_classifying_space} \linebreak
\noindent\hyperlink{_as_a_form_of_elliptic_cohomology}{$K(ku)$ as a form of elliptic cohomology}\dotfill \pageref*{_as_a_form_of_elliptic_cohomology} \linebreak
\noindent\hyperlink{from_gerbes_to_2vector_bundles}{From gerbes to 2-vector bundles}\dotfill \pageref*{from_gerbes_to_2vector_bundles} \linebreak
\noindent\hyperlink{2K-theory}{2K-theory of bimonoidal categories}\dotfill \pageref*{2K-theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{2vector_bundles_following_baasdundasrognes}{}\subsection*{{2-Vector bundles following Baas-Dundas-Rognes}}\label{2vector_bundles_following_baasdundasrognes}

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 \textbf{charted 2-vector bundle} (i.e. a [[cocycle]] for a BDR 2-vector bundle) of rank $n$ is

\begin{itemize}%
\item 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 bundle]]s $E^{\alpha \beta}_{i j} \to U_{\alpha \beta}$ such that the determinant of the underlying matrix of dimensions is $det(dim(E^{\alpha \beta}_{i j})) = \pm 1$.


\item on triple overlaps $U_{\alpha \beta \gamma}$ for $\alpha \lt \beta \lt \gamma \in A$ we have [[isomorphism]]s

\begin{displaymath}
\phi^{\alpha \beta \gamma}
  : 
  \oplus_{j} E^{\alpha \beta}_{i j} \otimes E^{\beta \gamma}_{j k}
    \stackrel{\simeq}{\to}
    E^{\alpha \gamma}_{i k}
\end{displaymath}

\item such that the $\phi$ satisfy on quadruple overlaps the evident cocycle condition (as described at [[gerbe]] and [[principal 2-bundle]]).



\end{itemize}
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.

\textbf{Remark} If we restrict attention to $n = 1$ then this gives the same as $U(1)$-gerbes/[[bundle gerbe]]s.

\textbf{Theorem} (Baas-Dundas-Rognes)

There exists a [[classifying space]] $\mathcal{K}(V)$ such that for $X$ a finite [[CW-complex]] there is an [[isomorphism]]

\begin{displaymath}
[X, \mathcal{K}(V)] = 
  {\lim_\to}_{a : Y \to X}
    Gr(2Vect(Y))
\end{displaymath}
between [[homotopy]] classes of continuous maps $X \to \mathcal{K}(V)$ and equivalence classes of the group completed [[infinity-stackification|stackification]] of 2-vector bundles,

where the [[colimit]] is over acyclic [[Serre fibration]]s (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.

\textbf{Proof} In BDR, Segal Birthday Proceedings

\textbf{Note} $2Vect_n(X) = [X, |B Gl_n (V)| ]$.

BDR called $\mathcal{K}(V)$ the \textbf{2-K-theory} of the [[bimonoidal category]] of [[Kapranov-Voevodsky 2-vector space]]s.

\hypertarget{the_homotopy_type_of_the_classifying_space}{}\subsection*{{The homotopy type of the classifying space}}\label{the_homotopy_type_of_the_classifying_space}

\textbf{Theorem} (Baas-Dundas-Rognes-Richter)

$\mathcal{K}(V) \simeq K(ku)$

Here:

\begin{itemize}%
\item $ku$ is the connected version of the [[spectrum]] of complex [[topological K-theory]];


\item $K(-)$ denotes forming the [[algebraic K-theory]] spectrum of [[ring spectrum|ring spectra]].



\end{itemize}
So by the general formula for [[algebraic K-theory]] for [[ring spectrum|ring spectra]], this is

\begin{displaymath}
K(ku) \simeq \mathbb{Z} \times B Gl(ku)^+
 \,
\end{displaymath}
Some flavor of $\mathcal{K}(V)$.

\begin{itemize}%
\item $V$ is the (a [[skeleton]] of the [[core]] of) of [[category]] of complex [[vector space]]s.

\begin{itemize}%
\item [[object]]s are [[natural number]]s, $n$ corresponding to $\mathbb{C}^n$;


\item [[morphism]]s are $Hom(k,k)  = GL(k)$ and there are no morphisms between different $k,l$.



\end{itemize}


\end{itemize}
This category $V$ is naturally a [[bimonoidal category]] under [[coproduct]] and [[tensor product]] of [[vector space]]s.

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

\begin{displaymath}
\mathcal{K}(V) = \Omega B (\coprod_{n \geq 0} B Gl_n (V))
  \,.
\end{displaymath}
Notice that this is a direct generalization of the corresponding formula for the algebraic K-theory of a ring $R$,

\begin{displaymath}
K(R) = \Omega B (\coprod_{n \geq 0} B Gl(R))
  \,.
\end{displaymath}
\hypertarget{_as_a_form_of_elliptic_cohomology}{}\subsection*{{$K(ku)$ as a form of elliptic cohomology}}\label{_as_a_form_of_elliptic_cohomology}

Ausoni and Rognes compute the [[homology]] groups (for a certain sense of homology) of $K(ku)$.

take rational homotopy

\begin{itemize}%
\item $H^*(-, \mathbb{Q})$

for $p$ a [[prime]], multiplying by $p$ gives an isomorphism on this.

p = $\nu_0$


\item Let $KU^*(-)$ be complex oriented [[topological K-theory]], then

\begin{displaymath}
KU_* = \mathbb{Z}[u^{\pm 1}]
\end{displaymath}
for $|u| = 2$ (the [[Bott class]]) we have that multiplying by $u$ is an isomorphism and $u^{p-1} = \nu_1$



\end{itemize}
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]].''

\hypertarget{from_gerbes_to_2vector_bundles}{}\subsection*{{From gerbes to 2-vector bundles}}\label{from_gerbes_to_2vector_bundles}

It is hard to directly construct charted 2-vector bundles. We have more examples of [[gerbe]]s. So we want to get one from the other.

\textbf{Example} We have

\begin{displaymath}
\mathbb{S}^1 = \mathbb{C}P^1 \subset \mathbb{C}P^\infty = B U(1) = K(\mathbb{Z},2)
\end{displaymath}
(see [[Eilenberg-MacLane space]] and [[classifying space]])

\begin{displaymath}
\mathbb{S}^3 = \Sigma \mathbb{S}^2 \to \Sigma B U(1) \subset 
  \Sigma B U \to B B U_{\otimes}
  \subset units(ku)
  \to B GL(ku)
\end{displaymath}
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

\begin{displaymath}
\mathbb{S}^3 \to K(\mathbb{Z},3)
\end{displaymath}
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 $\pi_3 K(\mathbb{Z},3) = H^3(\mathbb{S}^3)$

\textbf{Theorem} (Ausoni-Dundas-Rognes)

\begin{displaymath}
j(\mu) = 2 \zeta - \nu
\end{displaymath}
in $\pi_3(K(ku))$

so regarded as a 2-vector bundle $\mu$ is not a generator.

ADR: $\zeta$ is ``half a monopole''.

\begin{displaymath}
\pi_3(K(ku))  = \mathbb{Z} \oplus \mathbb{Z}/24 \mathbb{Z}
\end{displaymath}
(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.

\hypertarget{2K-theory}{}\subsection*{{2K-theory of bimonoidal categories}}\label{2K-theory}

Let $(R, \oplus, \otimes, 0,1, c_{\oplus})$ be a [[bimonoidal category]], i.e. a [[categorification|categorified]] [[rig]].

This can be broken down as

\begin{enumerate}%
\item $(R, \oplus, 0 , c_{\oplus})$ a permutative category, a categorified abelian [[monoid]];


\item $(R , \otimes, 1)$ is a [[monoidal category]], assumed to be \emph{strict} monidal in the following;


\item a distributivity law.



\end{enumerate}
\textbf{Examples}

\begin{enumerate}%
\item $E = Core$[[FinSet]], the [[core]] of the category of finite sets and morphisms only between sets of the same cardinality.

In the [[skeleton]], objects are [[natural number]]s $n \in \mathb{N}$, $\oplus$ and $\otimes$ is addition and multiplication on $\mathbb{N}$, respectively. Here $c_{\oplus}$ is the evident natural [[isomorphism]] between direct sums of finite sets.


\item $V = Core$[[Vect]] the [[core]] of the category of finite dim vector spaces, with morphisms only between those of the same dimension.


\item \ldots{}



\end{enumerate}
\textbf{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

\begin{displaymath}
\itexarray{
    Gl_n(R) &\hookrightarrow& Mat_n(R)
    \\
    \downarrow && \downarrow
    \\
    Gl_n(\pi_0(R))&\to& Mat_n(\pi_0(R))
    \\
    \downarrow && \downarrow
    \\
    Gl_n(Gr(\pi_0(R)))&\to& Mat_n(Gr(\pi_0(R)))
  }
  \,,
\end{displaymath}
where $Gr(-)$ denotes [[Grothendieck group]]-completion.

\textbf{Definition} (Baas-Dundas-Rognes, 2004)

For $R$ a [[bimonoidal category]], the \textbf{2K-theory} of $R$ is

\begin{displaymath}
\mathcal{K}(R) :=  \Omega B \coprod_{n \geq 0} | B Gl_n(R) |
\end{displaymath}
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

\begin{displaymath}
\cdots \simeq \mathbb{Z} \times |B Gl_n(R)|^+
  \,.
\end{displaymath}
Here $B_q Gl_n(R)$ is a [[simplicial category]]

\ldots{}

\textbf{Theorem} (Baas-Dundas-Rognes)

Let $R$ be a [[small category|small]] [[Top]]-[[enriched category|enriched]] [[bimonoidal category]] such that

\begin{enumerate}%
\item $R$ is a [[groupoid]];


\item for all $X \in R$ we have that $X \oplus (-)$ is [[faithful functor|faithful]].



\end{enumerate}
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).

\textbf{Examples}

\begin{enumerate}%
\item For the category $R := E = Core(FinSet)$ of finite sets as above we have that $H E$ is the [[sphere spectrum]].


\item For $R := V = Core(FinVect)$ the core of complex finite dimensional vector spaces we have $H V$ is the complex [[K-theory spectrum]].


\item For $V_{\mathbb{R}}$ analogously we get the real K-theory spectrum.



\end{enumerate}
So by the above theorem

\begin{enumerate}%
\item $\mathcal{K}(E) \simeq K(S) \simeq A(*)$


\item $\mathcal{K}(V) \simeq K(ku)$;


\item etc.



\end{enumerate}
\textbf{Remarks}

\begin{enumerate}%
\item A. Osono: The equivalence $\mathcal{K}(R) \simeq K(H R)$ of [[topological space]]s is even an equivalence of [[infinity loop space]]s;


\item Application of that:

a) for $E$ a [[ring spectrum]]: find a model

\begin{displaymath}
E \simeq H R(R)
\end{displaymath}
and use the equivalence $\mathbb{K}(R) \simeq K(H R E)$ to understand \emph{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 $\mathcal{K}(R)$.



\end{enumerate}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[red-shift conjecture]]

\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

The original articles on BDR 2-vector bundles are

\begin{itemize}%
\item [[Nils Baas]], [[Ian Dundas]], [[John Rognes]], \emph{Two-vector bundles and forms of elliptic cohomology}, in: \emph{Topology, geometry and quantum field theory}, volume 308 of London Math. Soc. Lecture Note Ser., pages 18--45. Cambridge Univ. Press, Cambridge, (2004).

\end{itemize}
Their [[classifying spaces]] are discussed in

\begin{itemize}%
\item [[Nils Baas]], [[Ian Dundas]], [[Birgit Richter]], [[John Rognes]], \emph{Ring completion of rig categories} (\href{http://arxiv.org/abs/0706.0531}{arXiv:0706.0531})

(a previous version of this carried the title \emph{Two-vector bundles define a form of elliptic cohomology})


\item [[Nils Baas]], M. B\"o{}ckstedt, [[Tore Kro]], \emph{Two-Categorical Bundles and Their Classifying Spaces} (2008) (\href{http://arxiv.org/abs/math/0612549}{arXiv:math/0612549}), .



\end{itemize}
Divisibility of the gerbe on the 3-sphere, seen as a 2-vector bundle is in

\begin{itemize}%
\item Christian Ausoni, Bjorn Ian Dundas and John Rognes, \emph{Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere}, Doc. Math. 13, 795-801 (2008). (\href{http://www.math.univ-paris13.fr/~ausoni/papers/ADR08.pdf}{pdf})

\end{itemize}
Orientation of BDR 2-vector bundles is discussed in

\begin{itemize}%
\item Thomas Kragh, \emph{Orientations and Connective Structures on 2-vector Bundles} Mathematica Scandinavica, 113 (2013) no 1, (\href{http://www.mscand.dk/article/view/15482}{journal}), (\href{http://arxiv.org/abs/0910.0131}{arXiv:0910.0131})

\end{itemize}
[[!redirects BDR 2-vector bundles]]



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