When do we study maps into an object or from the object to another object?

Question

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?

Answer 1

In combinatorics, one considers both maps out of a space $X$ (colourings) and maps into a space $X$ (tuples). But there is one key difference between the two: if $X$ has $n$ elements, then the number of maps from, say, $\{0,1\}$ to $X$ is polynomial in $n$ (it has order $n^2$), while the number of maps from $X$ to $\{0,1\}$ is exponential in $n$ (it has order $2^n$). Thus we see that maps out of $X$ into a simple space form a much larger, and presumably thus much richer, space than maps into $X$ from a simple space. For instance, deciding whether a four-colouring of a graph with certain specified properties exists is usually a harder problem than deciding whether a four-tuple in a graph with certain specified properties exists, although both questions can be interesting.

Of course, the situation could be different for other categories than the combinatorial one (particularly if some sort of duality is available)...

Answer 2

Let $V$ be a variety over a field $K$. In the category of varieties (or schemes) over $K$. I am very interested in studying all the morphisms from $\operatorname{Spec}(K)$ to $X$. I could hardly be less interested in studying the set of all morphisms from $X$ to $\operatorname{Spec}(K)$: this is, trivially, a single point.

On the other hand, if your variety $V$ is affine -- say $\operatorname{Spec} A$ -- then we are really saying that we prefer to study $K$-algebra maps from $A$ to $K$ (i.e., maps from $A$) rather than $K$-algebra maps from $K$ to $A$ (i.e., maps into $A$). This points to a curious feature of your question: it is probably most natural to construe it in terms of categories. But in this setup, if you just switch to the opposite category, the answer switches around!

Nevertheless I think your question is a real one. One could just as well ask: why do most categories come with a "natural orientation", i.e., why do we prefer the category to its opposite category? I think there's something to this question as well.

Answer 3

Here is a low-level observation, which I think I read on a different MO thread somewhere. If the objects in your category behave like the category of sets, then a map into an object $X$ can be "local" (its image might be a small subobject), but a map out of $X$ must always be "global" (it has to be defined on all of $X$). So in some sense even a single map out of $X$ (e.g. a Morse function in the category of smooth manifolds) can capture much more of the structure of $X$ than a single map into $X$.

Of course if your category behaves like the opposite of the category of sets then the opposite is true. And by the Yoneda lemma both maps into an object and maps out of an object classify it up to isomorphism, so I don't think it necessarily makes sense to privilege either point of view in general.

There is some really interesting general discussion of these issues in Lawvere and Schanuel's Conceptual Mathematics.

Answer 4

Here are two important generalizations of the notion of topological space:

1) C*-algebras.
Given a space X, one considers the set of continuous functions from X to ℂ. One then looks at all the properties of this set (it's an algebra, it's a Banach space, ...). One relaxes a bit the conditions (allow the multiplication to be non-commutative): there you get the notion of a C*-algebra.

2) Stacks.
Given a space X, one considers the collection of all maps Y→ X, where Y is an arbitrary space. We then look at at their properties (they form a category over Top¹, there exists a notion of precomposition with a map Y'→Y, they behave well w.r.t open covers, ...). Relax some conditions, and you'll get the notion of a stack.

By focusing on out of a space X, you get the notion of C*-algebra.
By focusing on maps into a space X, you get the notion of stack.

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¹Here, Top refers to the category of topological spaces. Stacks are encountered more often in algebraic geometry. In that case, one uses the category of schemes in place of Top.

Answer 5

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).