Grothendieck's relative point of view and Yoneda lemma

Question

I asked this question on M.SE, but didn't get any answers.

Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $X$ it's better to study morphisms into that object, $-\to X$. Apparently that's called the relative point of view.

First question. How is that principle applied in practice? What are some concrete examples in mathematics where the relative point of view is useful?

Wikipedia mentions the Riemann–Roch theorem and a similar MSE question mentions a theorem about coherent sheaves. Unfortunately, I don't know any algebraic geometry yet. Are there more down-to-earth applications of the relative point of view that an undergraduate can understand, say, in linear algebra, group theory, ring theory, Galois theory, or maybe even in basic category theory?

What are (some of) the most important theorems that feature the relative point of view?

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I recently heard about the Yoneda lemma in category theory (I know the statement and can prove it). I know that it can be used to prove that two objects are isomorphic whenever they have the same universal property. In Awodey's category theory book, there's a concrete application of that: in categories with enough structure, $(A\times B)+(A\times C)\cong A\times (B+C)$. That proof is elegant, I agree. But it doesn't live up with the praise many people give to the Yoneda lemma, does it?

Maybe a more concrete application in non-category theory would help me to get convinced of the contrary. For instance, I read on Wikipedia (and elsewhere) that Grothendieck used the Yoneda lemma in his famous book EGA (which a lot of people seem to talk about). (In fact, it seems this was another insight of him: that Yoneda is useful.)

Second question. So what were Grothendieck's main applications of the Yoneda lemma in algebraic geometry? (In contrast to the first question, here it suffices for me to just know roughly what kind of statement he proved with the Yoneda lemma---rather than understanding it in detail, because I already know one application of the Yoneda lemma.)

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Third question. Is the second question related to the first one, i.e., is there a connection between the relative point of view and the Yoneda lemma? (At least the Wikipedia page linked above mentions the Yoneda lemma.)

Answer 1

Let me answer your questions in reverse order.

For the last question, yes, Yoneda's lemma is absolutely crucial to the relative point of view, as it essentially postulates that passing from a scheme $X$ (or more generally an object of any category) to the Hom functor $Hom(-,X)$ does not lose information. More precisely, the functor $Hom(-,X)$ determines the object $X$ up to isomorphism, and moreover natural transformations between functors $Hom(-,X)$ and $Hom(-,Y)$ are in correspondence with morphisms from $X$ to $Y$ (so you recover not only objects, but also morphisms). In the case of schemes, it turns out that you don't need to consider $Hom(Z,X)$ for all schemes $Z$, it suffices to consider them for $Z=\operatorname{Spec}R$ for a ring $R$ - in this case, $Hom(Z,X)$ should be thought of as the set of points of $X$ with coefficients in $R$. This is not something that follows automatically from categorical considerations, but rather to the fact schemes are "locally affine".

As for the applications of Yoneda, one point is that in general describing a morphism of schemes directly can be rather annoying - working at the level of underlying topological spaces and structure sheaves would be quite tedious. Fortunately, using the remarks of my previous paragraph, to specify a morphism it is enough to specify maps at the level of $R$-valued points for all rings $R$ (in a functorial manner). A good example of this is given by group schemes - for instance, if you have an elliptic curve $E$, then for any ring you can define a group structure on the set $E(R)$ using the formulas which are not hard to derive from the geometric construction of the addition law. This gives a function from $E(R)\times E(R)\cong(E\times E)(R)$ to $E(R)$, and induces a morphism $E\times E\to E$.

Talking about any of the Grothendieck's results won't be easy if you are not familiar with algebraic geometry, but let me sketch the idea behind one such, namely the theory of the Picard functor. To any (smooth projective) algebraic curve $C$ over a field, one can construct another variety $J(C)$, known as its Jacobian, which parametrizes divisors (linear combinations of points) on $C$ modulo a certain equivalence relation. There are many ways to construct it, but relative point of view gives a nice explanation of what it "is". Specifically, for any other variety $V$, we can consider "families of divisors on $C$ indexed by $V$" - those are certain ("nice") divisors on $V\times C$. Jacobian $J(C)$ then has the following property: such families of divisors indexed by $V$ are completely determined by maps $V\to J(C)$. We say that $J(C)$ represents the functor taking a variety $V$ to the set of all such families of divisors. This is something that only makes sense once we consider a relative point of view, generalizing the curve $C$ to the "family of curves" $V\times C\to V$.

In general, Grothendieck has described an analogous functor for any "relative curve" over another scheme; in particular one can consider such over arbitrary ring (not just a field) and has shown that such functors are always representable, by a scheme called the Picard scheme. This is again something we couldn't even make sense of without considering the relative point of view.

Answer 2

There is a crucial aspect of the relative point of view that I think has not been completely covered here so far.

The relative point of view does not just mean that we want to look at morphisms. It says we want to look at morphisms as analogues of objects.

How does this analogy work? A space $X$ (for example a topological space, manifold, or scheme) has the same information as a map from $X$ to a point. So a map between two spaces is a generalization of a space, which we can think of as a family of spaces, the fibers, parameterized by the points of the target.

Families of spaces were certainly something people considered before Grothendieck, but a big part of the philosophy is to think of every morphism as a family of this type, even ones where the fibers are very different from each other.

One thing this means is that for every interesting property that a space can satisfy, there should be an analogous property that a morphism can satisfy. The relative point of view tells you to look for this, and to generalize theorems about the old property to theorems about the new.

There's many examples of this in scheme theory, but a more elementary field where the same insight applies is point-set topology.

For example, compact topological spaces are interesting. What could the generalization to morphisms be? It must be a property of maps $X \to Y$ of topological spaces which are satisfied when $Y$ is a point if and only if $X$ is compact. One can look at maps where the inverse image of every compact set is compact (which are at least sometimes called proper maps), or universally closed maps, i.e. maps where for every topological space $Z$, the image of every closed subset of $X \times Z$ inside $Y \times Z$ is also closed.

Answer 3

Another statement of the relative perspective is that one should consider morphisms $X\to B$ of schemes as families of schemes (i.e. all the fibers) indexed by the base $B$. We then use this idea to extend properties of individual schemes to properties of families of schemes, i.e. properties of morphisms. Similarly, one can identify sheaves on a product $X\times B$ with families of sheaves on $X$ indexed by $B$.

This connects with the Yoneda lemma through the functor of points perspective; we view a scheme $X$ as the functor it represents. Indeed, many constructions in algebraic geometry are given by specifying the functor of points of a scheme we want to construct, and these functors generally have the form $T\mapsto \{\text{some kind of families indexed by }T\}$. For example, we have the Jacobian of a curve $C$ mentioned above, whose functor of points (roughly) is given by $T\mapsto \{\text{families of degree zero line bundles on }C\text{, indexed by T}\}$, and the Hilbert scheme of a scheme $X$, roughly given by $T\mapsto \{\text{families of closed subschemes of }X\text{, indexed by }T\}$. Here, the power of the relative perspective as formulated above is in formalizing the notion of families indexed by a base.