Parametrized families in a categorical language Let $X$ be a set, and for each $x\in X$ let $Y_x$ be a complex vector space.
Is there a standard and broad language for such a thing in terms of the categories of sets and of complex vector spaces? It seems like the idea of a set-theoretic function from an object in one category to the set of objects in another category is a little inelegant.
 A: There are several ways to think about these kinds of things. Another good one is via fibered categories. There is a functor $T:\mathcal E\to \mathrm{Set},$ where objects of $\mathcal E$ are pairs of a set $X$ and an $X$-indexed family of vector spaces. For any function $f:X\to Y,$ there is a pullback operation sending a family $V=(V_y)_{y\in Y}$ to the family $f^*V=(V_{f(x)})_{x\in X},$ and generally, morphisms from $(V,X)$ to $(U,Y)$ are given by choosing some $f:X\to Y$ and a morphism $V\to f^*U,$ where the definition of morphism between two families of vector spaces over $X$ is probably what you think it is.
With all that structure in place, the functor $T$ (which simply sends $(V,X)$ to $X$) is of a special kind, called a Grothendieck fibration. A fibered category is the domain of a Grothendieck fibration. Generally, a Grothendieck fibration can be identified with a weak type of functor from its codomain into the 2-category of categories, sending each $X$ in this case to the category of $X$-indexed families of vector spaces.
This may not be the kind of answer you were hoping for, since you found the idea of a function from a set to a set (class, really) of objects, inelegant. If so, well, that's really a set-theoretic issue anyway. Category theory is about categories, not about the concrete nature of their objects.
That said, this does move your focus in a productive direction, to looking at the categorical relationships between the different categories of $X$-indexed vector spaces. Grothendieck fibrations can also be generalized very far beyond this example, and they have an extremely rich theory. For instance, Grothendieck fibrations over an arbitrary topos generalize locally small categories very effectively--one can study their limits and colimits by asking for adjoints to functors like $f^*$, and then completeness properties, etc, etc. So there is indeed a standard, broad, and powerful language into which your example fits.
