I have some understanding that vector bundles provide a basic, familiar example of what I should call a stack. Namely, consider the functor $Vect$ that assigns to a space $X$ the *set* of isomorphism classes of vector bundles on $X$. This functor isn't local, in the sense that the isoclass of a vector bundle isn't determined by its restriction to an open cover, but rather by *gluing data* on overlapping sets in a cover. Since for any space $Y$ a map $X \to Y$ *is* determined by what it does when restricted to a cover of $X$, this tells us there is no space $Y$ that represents the functor $Vect$ in this fashion. However, I can also consider $Vect$ as a stack, which assigns to $X$ the *groupoid* of vector bundles on $X$. This gadget is fancy enough to understand how vector bundles glue together, and so recovers the locality missing from our earlier functor.

In K-theory, we attach to a space $X$ a ring $K(X)$ whose underlying group is the the free abelian group on the set of isoclasses of vector bundles on $X$, mod short exact sequences. It turns out that one can describe $K(X)$ as the set of homotopy classes of maps from $X$ to $\mathbb{Z} \times BU(\infty)$.

At this point my meager understanding of K-theory seems to be contradict what I said in the first paragraph. The fact that $K(X)$ has a classifying space seems at odds with the observation that vector bundles aren't determined by their restrictions to open covers, whereas maps to another space are. Is something wrong with what I've said so far? If not, perhaps there isn't a contradiction because either 1) $K(X)$ isn't quite the set of isoclasses of vector bundles, but rather a group completion thereof, or 2) we're looking at *homotopy classes* of maps to $\mathbb{Z} \times BU(\infty)$, so what I said in the first paragraph doesn't apply?