Is there a notion of "space" such that vector bundles can be understood in this way? Is there a notion of "space" satisfying the following requirements?


*

*Spaces form (at least) a category; morphisms between spaces are called "continuous maps."

*Every topological space is a space, and the inclusion of $\mathbf{Top}$ in the category of spaces is fully faithful and injective on objects.

*The category $\mathbb{R}\mathbf{Mod}$ is a space.

*If $X$ is a topological space, a vector bundle over $X$ can be described as a continuous map $X \rightarrow \mathbb{R}\mathbf{Mod}.$
(I don't mind if some of these requirements are partially violated, they're just meant to be guidelines.)
 A: As Qiaochu said, probably what you want are topological stacks.


*

*Let $T$ be a small full subcategory of $\mathrm{Top}$, with the Grothendieck topology of open covers, and consider the 2-category of stacks of groupoids on $T$.  Call its objects "spaces" and its morphisms "continuous maps".  You could call its 2-cells "continuous transformations".

*Any topological space $X$ determines a sheaf $\mathrm{Top}(-,X)$ on $T$ and hence a stack.  This functor is fully faithful on $T$, and often on a much larger subcategory of $\mathrm{Top}$.  (For instance, if $T=\{\mathbb{R}^n\}$ then the functor is fully faithful on at least all topological manifolds.)

*The functor $T^{op} \to \mathrm{Gpd}$ defined by sending $X\in T$ to the groupoid of real vector bundles over $T$ is a stack, because vector bundles can be glued together over open covers.  Call it your $\mathbb{R}\mathbf{Mod}$.

*If $X\in T$ (and often for many more $X\in\mathrm{Top}$), then by the Yoneda lemma, the groupoid of continuous maps and continuous transformations $\mathrm{Top}(-,X) \to \mathbb{R}\mathbf{Mod}$ is equivalent to $\mathbb{R}\mathbf{Mod}(X)$, i.e. the groupoid of real vector bundles on $X$.
A: NB As Qiaochu Yuan explains in the comment below, what follows is not correct: it only captures a very drastic quotient of the isomorphism groupoid of vector bundles; most likely - the groupoid of connected components of homs of the latter.
In a sense, there is a way to stay inside $\mathbf{Top}$. If I am not mistaken, the groupoid $[\![X,\coprod_nBO(n)]\!]$ with objects continuous maps $X\to\coprod_nBO(n)$ and morphisms homotopy classes of homotopies is equivalent to the groupoid of vector bundles on $X$ and their isomorphisms.
I don't know how to recover in this way vector bundle morphisms that are not isomorphisms, but OP does not mention this, so... :P
