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?



Is there a categorification of topological K-theory?
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.


