Is there a categorification of topological K-theory? For a compact Hausdorff topological space $X$, its K-theory $K^0(X)$ is defined to be the Grothendieck group of the isomorphism classes of finite dimensional vector spaces on $X$. For example $K^0(\text{pt})=\mathbb{Z}$.
My question is: is there a categorification of K-theory, just as we can categofify natural numbers to vector spaces?
 A: One answer these days is to think of $K$-theory as represented by a commutative ring spectrum
(alias $E_{\infty}$-ring spectrum) $K$.  Then there is a perfectly good theory of $K$-module
spectra, to which we can apply Waldhausen's approach to algebraic $K$-theory to construct the
$K$-theory of $K$-theory.  This first appeared in EKMM (Elmendorf-Kriz-Mandell-May. Rings, modules,
and algebras in stable homotopy theory.  AMS 1997) and has been much studied since, for example by Blumberg and Mandell.  The localization sequence for the algebraic $K$-theory of topological $K$-theory.
A: The abelian group $K^0(X)$ has a natural $(\infty,0)$-categorification'', meaning a spectrum $K(X)$ whose $\pi_0$ recovers $K^0(X)$: take $K(X)$ to be the spectrum of maps from (the suspension spectrum of) $X$ to the (complex or real, depending) K-theory spectrum.  Or we could try a different variant, the spectrum given as the group completion of the topological groupoid of vector bundles over X under direct sum; I think this variant would give the connective cover of $K(X)$.
To be clear, this is an analog of natural numbers categorifying to finite-dimensional vector spaces only if you force yourself to neglect the non-invertible maps between vector spaces.
A: Maybe I am missing something, but is there a reason no one has mentioned the work on 2-vector spaces. One place to start is http://hopf.math.purdue.edu//Baas-Dundas-Rognes/segal60.pdf . They state a conjecture that the algebraic K-theory of the category of 2-vector spaces (over $\mathbb{C}$) is the algebraic K-theory of $ku$. The conjecture is proved in http://arxiv.org/pdf/0706.0531.pdf by Baas, Dundas, Richter and Rognes.
