nLab
BDR 2-vector bundle

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.

Contents

2-Vector bundles following Baas-Dundas-Rognes

We are looking for a generalization of the notion of vector bundle in higher category theory.

Let XX be a topological space and {U αX} αA\{U_\alpha \to X\}_{\alpha \in A} an open cover, where the index set AA is assumed to be a poset.

Defition. A charted 2-vector bundle (i.e. a cocycle for a BDR 2-vector bundle) of rank nn is

  • for α<βA\alpha \lt \beta \in A on U αU β=:U αβU_\alpha \cap U_\beta =: U_{\alpha\beta} a matrix (E ij αβ) i,j=1 n(E^{\alpha\beta}_{i j})_{i,j = 1}^{n} of vector bundles E ij αβU αβE^{\alpha \beta}_{i j} \to U_{\alpha \beta} such that the determinant of the underlying matrix of dimensions is det(dim(E ij αβ))=±1det(dim(E^{\alpha \beta}_{i j})) = \pm 1.

  • on triple overlaps U αβγU_{\alpha \beta \gamma} for α<β<γA\alpha \lt \beta \lt \gamma \in A we have isomorphisms

    ϕ αβγ: jE ij αβE jk βγE ik αγ \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}
  • such that the ϕ\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)2Vect(X) for the equivalence classes of charted 2-vector bundles under these morphisms.

Remark If we restrict attention to n=1n = 1 then this gives the same as U(1)U(1)-gerbes/bundle gerbes.

Theorem (Baas-Dundas-Rognes)

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

[X,𝒦(V)]=lim a:YXGr(2Vect(Y)) [X, \mathcal{K}(V)] = {\lim_\to}_{a : Y \to X} Gr(2Vect(Y))

between homotopy classes of continuous maps X𝒦(V)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()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,|BGl n(V)|]2Vect_n(X) = [X, |B Gl_n (V)| ].

BDR called 𝒦(V)\mathcal{K}(V) the 2-K-theory of the bimonoidal category of Kapranov-Voevodsky 2-vector spaces.

The homotopy type of the classifying space

Theorem (Baas-Dundas-Rognes-Richter)

𝒦(V)K(ku)\mathcal{K}(V) \simeq K(ku)

Here:

So by the general formula for algebraic K-theory for ring spectra, this is

K(ku)×BGl(ku) + K(ku) \simeq \mathbb{Z} \times B Gl(ku)^+ \,

Some flavor of 𝒦(V)\mathcal{K}(V).

This category VV 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)Mat_n(V) for the n×nn \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)Gl_n(V) for the subcollection of those matrices for which the determinant of their matrix of dimensions is ±1\pm 1.

Now define

𝒦(V)=ΩB( n0BGl n(V)). \mathcal{K}(V) = \Omega B (\coprod_{n \geq 0} B Gl_n (V)) \,.

Notice that this is a direct generalization of the corresponding formula for the algebraic K-theory of a ring RR,

K(R)=ΩB( n0BGl(R)). K(R) = \Omega B (\coprod_{n \geq 0} B Gl(R)) \,.

K(ku)K(ku) as a form of elliptic cohomology

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

take rational homotopy

  • H *(,)H^*(-, \mathbb{Q})

    for pp a prime, multiplying by pp gives an isomorphism on this.

    p = ν 0\nu_0

  • Let KU *()KU^*(-) be complex oriented topological K-theory, then

    KU *=[u ±1] KU_* = \mathbb{Z}[u^{\pm 1}]

    for |u|=2|u| = 2 (the Bott class) we have that multiplying by uu is an isomorphism and u p1=ν 1u^{p-1} = \nu_1

The ν i\nu_i come from the Brown-Peterson spectrum BPB P and π *BP= (p)[ν 1,ν 2,]\pi_* BP = \mathbb{Z}_{(p)}[\nu_1, \nu_2, \cdots]

motto: the higher the ν i\nu_i the more you detect.

Christian Ausoni figured out something that implies that

K(ku)K(ku) 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

𝕊 1=P 1P =BU(1)=K(,2) \mathbb{S}^1 = \mathbb{C}P^1 \subset \mathbb{C}P^\infty = B U(1) = K(\mathbb{Z},2)

(see Eilenberg-MacLane space and classifying space)

𝕊 3=Σ𝕊 2ΣBU(1)ΣBUBBU units(ku)BGL(ku) \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)

using ΣBU(1)BBU(1)BBU \Sigma B U(1) \to B BU(1) \to B B U_{\otimes} we can take a U(1)U(1)-gerbe classified by maps into B 2U(1)B^2 U(1) and induce from it the associated 2-vector bundle.

the canonical map

𝕊 3K(,3) \mathbb{S}^3 \to K(\mathbb{Z},3)

may be thought of as classifying the gerbe called the magnetic monopole-gerbe

Postcomposing with μ:K(,3)K(ku)\mu : K(\mathbb{Z},3) \to K(ku) we have

Fact: μ\mu gives a generator in π 3K(,3)=H 3(𝕊 3)\pi_3 K(\mathbb{Z},3) = H^3(\mathbb{S}^3)

Theorem (Ausoni-Dundas-Rognes)

j(μ)=2ζν j(\mu) = 2 \zeta - \nu

in π 3(K(ku))\pi_3(K(ku))

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

ADR: ζ\zeta is “half a monopole”.

π 3(K(ku))=/24 \pi_3(K(ku)) = \mathbb{Z} \oplus \mathbb{Z}/24 \mathbb{Z}

(the first summand is ζ\zeta, the second ν\nu).

Thomas Krogh has an orientation theory for 2-vector bundles which says that j(ν)j(\nu) is not orientable.

2K-theory of bimonoidal categories

Let (R,,,0,1,c )(R, \oplus, \otimes, 0,1, c_{\oplus}) be a bimonoidal category, i.e. a categorified rig.

This can be broken down as

  1. (R,,0,c )(R, \oplus, 0 , c_{\oplus}) a permutative category, a categorified abelian monoid;

  2. (R,,1)(R , \otimes, 1) is a monoidal category, assumed to be strict monidal in the following;

  3. a distributivity law.

Examples

  1. E=CoreE = CoreFinSet, the core of the category of finite sets and morphisms only between sets of the same cardinality.

    In the skeleton, objects are natural numbers nmathbNn \in \mathb{N}, \oplus and \otimes is addition and multiplication on \mathbb{N}, respectively. Here c c_{\oplus} is the evident natural isomorphism between direct sums of finite sets.

  2. V=CoreV = CoreVect the core of the category of finite dim vector spaces, with morphisms only between those of the same dimension.

Definition For RR a bimonoidal category, write Mat n(R)Mat_n(R) for the n×nn \times n matrices with entries morphisms in RR. Then matrix multiplication is defined using the bimonoidal structure on RR. This gives a weak monoid structure.

Let Gl n(R)Gl_n(R) be the category of weakly invertible such matrices. This is the full subcategory of Mat n(R)Mat_n(R). We get a diagram of pullback squares

Gl n(R) Mat n(R) Gl n(π 0(R)) Mat n(π 0(R)) Gl n(Gr(π 0(R))) Mat n(Gr(π 0(R))), \array{ 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))) } \,,

where Gr()Gr(-) denotes Grothendieck group-completion.

Definition (Baas-Dundas-Rognes, 2004)

For RR a bimonoidal category, the 2K-theory of RR is

𝒦(R):=ΩB n0|BGl n(R)| \mathcal{K}(R) := \Omega B \coprod_{n \geq 0} | B Gl_n(R) |

where the Ω\Omega is forming loop space, the leftmost BB is forming classifying space of a category and the inner BB is a flabby version of classifying space of a category.

This can also be written

×|BGl n(R)| +. \cdots \simeq \mathbb{Z} \times |B Gl_n(R)|^+ \,.

Here B qGl n(R)B_q Gl_n(R) is a simplicial category

Theorem (Baas-Dundas-Rognes)

Let RR be a small Top-enriched bimonoidal category such that

  1. RR is a groupoid;

  2. for all XRX \in R we have that X()X \oplus (-) is faithful.

Then 𝒦(R)K(HR)\mathcal{K}(R) \simeq K(H R) is the ordinary algebraic K-theory of the ring spectrum HRH R.

Notice that for HRH R to be a spectrum we only need the additive structure (R,,0,c )(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

  1. For the category R:=E=Core(FinSet)R := E = Core(FinSet) of finite sets as above we have that HEH E is the sphere spectrum.

  2. For R:=V=Core(FinVect)R := V = Core(FinVect) the core of complex finite dimensional vector spaces we have HVH V is the complex K-theory spectrum.

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

So by the above theorem

  1. 𝒦(E)K(S)A(*)\mathcal{K}(E) \simeq K(S) \simeq A(*)

  2. 𝒦(V)K(ku)\mathcal{K}(V) \simeq K(ku);

  3. etc.

Remarks

  1. A. Osono: The equivalence 𝒦(R)K(HR)\mathcal{K}(R) \simeq K(H R) of topological spaces is even an equivalence of infinity loop space?s;

  2. Application of that:

    a) for EE a ring spectrum: find a model

    EHR(R) E \simeq H R(R)

    and use the equivalence 𝕂(R)K(HRE)\mathbb{K}(R) \simeq K(H R E) to understand arithmetic properties of EE.

    b) Often one knows K(H(R))K(H(R)) via calculations. here 𝒦(R)\mathcal{K}(R) might help to get some deeper understanding.

    c) Theorem (Birgit Richter): for RR a bimonoidal category with anti-involution, then you get an involution of 𝒦(R)\mathcal{K}(R).

References

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

Divisibility of the gerbe on the 3-sphere, seen as a 2-vector bundle is in

  • Christian Ausoni, Bjorn Ian Dundas and John Rognes, Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere, Doc. Math. 13, 795-801 (2008). (pdf)

Orientation of BDR 2-vector bundles is discussed in

  • Thomas Kragh, Orientations and Connective Structures on 2-vector Bundles Mathematica Scandinavica, 113 (2013) no 1, (journal), (arXiv:0910.0131)

Revised on December 3, 2014 00:49:03 by David Roberts (129.127.211.9)