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?

algebraicK-theory, which he calls secondary K-theory. I have a set of notes (math.uchicago.edu/~ejenkins/nwtft.html) from lectures he gave a few years ago. (Apologies for my poor handwriting.) $\endgroup$ – Evan Jenkins Aug 4 '12 at 20:59