# n-categorical description of Chern classes

The Chern classes of a rank $n$ vector bundle on $X$ are obtained from composing the associated classifying map in $[X, BU(n)]$ with the maps $BU(n) \to B^{2i} \mathbb{Z}$ corresponding to the cohomology of $BU(n)$, thus giving elements in the cohomology groups $H^{2i}(X,\mathbb{Z})$.

But one could instead write those maps as $BU(n) \to B^{2i-1} U(1)$; and this corresponds to associating a $(2i-1)$-circle bundle to a vector bundle. For $i = 1$, for instance, this corresponds to sending a vector bundle to its top exterior power, the determinant line bundle.

What is then the interpretation of the higher Chern classes in this geometric setting? These should associate to the original rank $n$ vector bundle some circle $(2i-1)$-bundles (e.g. principal $B^{2i-2}U(1)$ $(2i-1)$-bundles), for $1 \leqslant i \leqslant n$.
This should mirror the algebraic side with symmetric polynomials; using $e_1 = x_1 + \ldots + x_n$ we have the line bundle $L = L_1 \otimes \ldots \otimes L_n$ (corresponding to the Chern roots); for other elementary symmetric polynomials $e_i$ we should be able to build a corresponding $(2i-1)$-line bundle directly realising the $i$-th Chern class. (Perhaps it would be more natural to instead consider Schur classes, and hopefully the relationship of these $(2i-1)$-line bundles with Schur functors is elucidated.)
Is there then an obstruction-theoretic picture, when trying to build back up the original vector bundle from these successive $(2i-1)$-line bundles?

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Great question! I would love to see a conceptual solution. However I think there's an obstacle to a robustly geometric solution (though any integral cohomology class can be reinterpreted as a "higher line bundle"). Namely in algebraic geometry Chern classes are of type $(p,p)$, i.e., $H^p(X,\Omega^p)$) while higher line bundles" or n-gerbes are of type $(0,n)$, i.e., $H^{n}(X,O^{\times})$. To see algebraic Chern classes along your lines we need to replace $O^{\times}=K_1$ by algebraic K-theory $K_n$, which is of unclear intuitive utility even for $n=2$..but maybe in topology all is well! – David Ben-Zvi Dec 28 '12 at 3:14
Rather than a $(2i-1)$-bundle over $BU(n)$, I'd look for a $U(n)$-equivariant structure on the (trivial) $(2i-1)$-bundle over the point. This is also known as a multiplicative $(2i-2)$-bundle over $U(n)$. For $i=2$, this is a multiplicative gerbe over $U(n)$. To a $U(n)$ vector bundle one can then associate a 2-gerbe, called the Chern-Simons 2-gerbe, which represents the corresponding class in $H^{2i}(X,\mathbb{Z})$. – Konrad Waldorf Dec 28 '12 at 13:38
What on earth does this question have to do with n-categories??? – John Pardon Dec 28 '12 at 14:31
@unknown (google): an $n$-bundle is an object in an $n$-category. This is relevant because, as mentioned in my earlier comment, one needs to consider additional structures like equivariant or multiplicative structure. Those make use of the higher categorical setting. – Konrad Waldorf Dec 28 '12 at 16:00
@David Ben-Zvi: I don't think your objection is too serious. First Chern classes of algebraic line bundles land in $H^{1,1}(X,\mathbb{Z}) \subseteq H^2(X, \mathbb{Z})$; we don't expect this to classify algebraic line bundles up to algebraic equivalence. Here it'll be similar: $H^{2i}(X,\mathbb{Z})$ classifies (topological) principal $B^{2i-2} \mathbb{G}_\mathrm{m}$ $(2i-1)$-bundles, and $H^{i,i}(X,\mathbb{Z})$ inside that corresponds to those bundles which admit an algebraic structure. – Will Jan 1 '13 at 11:12

The answer will depend on your realisation of a $k$-circle bundle. In the case of $i=2$ (second Chern class) there are results associating to any principal $G$-bundle a bundle $2$-gerbe. See:

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories. Alan L. Carey, Stuart Johnson, Michael K. Murray, Danny Stevenson and Bai-Ling Wang. Communications in Mathematical Physics, 159 (3) (2005), 577-613 math.DG/0410013

and the references there in to Danny Stevenson and Stuart Johnson's PhD theses and papers. Of course you have to be happy that a $3$-circle bundle is a $2$-gerbe.

More generally you might find something useful in:

P. Gajer Geometry of Deligne cohomology, Invent. Math. 127 (1997), 155-207.

which gives a realisation of principal $B^k \mathbb{C}^*$ bundles which are another possible way of realising $(k+1)$-circle bundles or at least mathematical objects determined by a characteristic class in degree $H^{k+1}(M, \mathbb{Z})$. There is a nice inductive classifying theory and a simplicial realisation of these spaces.

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Gajer's $B^k\mathbb{C}^{*}$-bundles will probably not work, since they do not capture the correct higher categorical properties, and these are essential to consider e.g. a multiplicative structure. Bundle $n$-gerbes are better :-) – Konrad Waldorf Dec 28 '12 at 16:11