TITLE: When do we study maps into an object or from the object to another object?
QUESTION [5 upvotes]: In many Mathematical theories, to study an object, we usually consider the set of all maps from that object to some other object. For example, in differential geometry, we study the smooth maps from a manifold $M$ to $\mathbb{R}$. Or in Algebraic Geometry, we consider the structure sheaf, which is the set of maps from a variety to $\mathbb{A}^1$.
So, is there any heuristic idea about why we don't do the other way around, i.e. study the set of all maps from some object to the object we want to study (at least in the two examples above)? Would this give us any more information? And also, is there any subject in which we do that?
Edit: One more clarification that might make my question clearer. In algebraic geometry, when we write an $R-$scheme $\mathrm{Spec} A$, already implicitly, we are viewing $A$ as the ring of all $R-$functions from $\mathrm{Spec} A$ to $\mathrm{Spec} R[x]$.
Edit (based on Qiaochu Yuan's answer): maps in seem to give us local information while maps out gives us global one, at least in Differential Geometry and Algebraic geometry. For example, to learn about the tangent space at a point, we look at the map $I\to M$ (in differential geometry) and $\mathrm{Spec} k[x]/(x^2) \to X$ (in algebraic geometry). Is there any more example along these lines?

REPLY [2 votes]: To me, it seems that your question is essentially "Why does contravariance occur more frequently than covariance?"
If one has a contravariant representable functor, then one is implicitly studying maps into a fixed (universal) object from the object one is interested in studying.
I think perhaps one reason why contravariance is more natural than covariance is what Qiaochu indicates in his answer: contravariant functors have more of an opportunity to be "local." For instance, let $X, Y$ be topological spaces, and $f: X \to Y$ a continuous map. Then a sheaf on $Y$ when pulled back to $X$ at a point $x$ depends only on the local nature near the single point $f(x)$. Things are not so nice when pushing forward a sheaf.  Thus it happens that pull-backs preserve stalks, while push-forwards need not (unless you are working with a particularly nice map $f$).
In particular, if one has a bundle on $Y$, the local nature implies that it can be pulled back to $X$, while pushing a vector bundle forward will only give a sheaf, not necessarily a locally free one (i.e. a bundle).
In algebraic geometry, one of the first representable functors one encounters is the one that represents projective space. Namely, fix a field $k$ and an integer $n$; then a map from a $k$-scheme $X$ into $\mathbb{P}^n_k$ is given by a  line bundle on $X$ and $n+1$ global sections generating it (up to isomorphism). This is contravariant because you can pull-back line bundles and the generating property of global sections. You can't push this forward in a reasonable manner. (This universal property generalizes to projective space bundles over any scheme.)
At an even more basic level, the definition of a sheaf itself is contravariant (it's a contravariant functor from the category of open sets with inclusions to the category of sets that satisfies unique gluing axioms).