Contents

Yoneda lemma

# Contents

## Idea

The rough idea is (but see the caveat below) that the term moduli space is essentially a synonym for representing object and for classifying space. People tend to say “classifying space” when in the context of topology, and they tend to say moduli space when in a context of complex geometry or algebraic geometry.

More precisely, when a moduli space actually does exist as an ordinary space (or scheme), it is called for emphasis a fine moduli space. Fine here refers to the completeness of the description, not shared by coarse moduli below.

Those classifying “spaces” that are called moduli spaces are typically orbifolds, hence modeled not just as spaces but as groupoids with extra structure. These are typically conceived as stacks: these are then called moduli stacks. Typically these are demanded to be Deligne-Mumford stacks.

So the term fine moduli space mainly indicates that a given object that might be a Deligne-Mumford stack is actually just a plain scheme. But there is also the notion of coarse moduli space, which is a kind of conceptual hack designed to be able to keep thinking about what really wants to be a stack still as a plain sheaf.

A coarse moduli space is one that at least has the right underlying set of points as the right moduli stack has: as long as we don’t look at families but just at single things, it does give the right information. From the point of view of derived algebraic geometry, the coarse moduli spaces are 0-truncations of derived moduli stacks when they exist.

### Caveat

But one has to be careful with this rough idea: as there is usually some implicit fine print in the notion of moduli space:

while classifying space is the term typically used for a representing object in a homotopy category, a representing object is usually called a moduli space only if it lives in the “original” category. For more on this see the chapter

below.

## History

The term possibly originates with Riemann, who was the first to study what are now called moduli spaces of (compact) Riemann surfaces. A “modulus” here is meant to be a parameter that parameterizes isomorphism classes of Riemann surfaces.

## Definition

### Coarse moduli space

A coarse moduli space for a presheaf $F:(Sch/\mathbb{C})^{op} \to Set$ on complex schemes is a scheme $M \in Sch/\mathbb{C}$ equipped with a morphism

$\Psi_M : F \to h_M \,,$

where $h$ denotes the Yoneda embedding, such that

a) $F(Spec(\mathbb{C})) \to h_M(Spec \mathbb{C}) = hom(Spec \mathbb{C}, M)$ is a bijection

b) given $M'$ and $\Psi_{M'} : F \to h_{M'}$ then there exists unique $M \to M'$ such that $\array{ F && \stackrel{\Psi_{M'}}{\to}&& h_{M'} \\ & {}_{\Psi_M}\searrow && \nearrow \\ && h_M}$

So a coarse moduli space is one that at least has the right underlying set of points as the right moduli stack has: as long as we don’t look at families but just at single things, it does give the right information. From the point of view of derived algebraic geometry, the coarse moduli spaces are 0-truncations of derived moduli stacks when they exist.

## “… because they have automorphisms.”

Slogan: A type of objects that has nontrivial automorphisms cannot have a fine moduli space.

The typical example motivating and illustrating this slogan is the example of elliptic curves: from a single elliptic curve $E$ with a non-trivial automorphism $\alpha : E \to E$ one may build, for instance, the family $\mathcal{E} \to S^1 \times S^1$ of elliptic curves over the torus which is obtained by starting with the trivial family $E \times S^1 \times [0,1] \to S^1 \times [0,1]$ over the cylinder and then by gluing the two ends by identifying the fiber using the automorphism $\alpha$. The resulting family is locally trivial but globally nontrivial.

Such locally trivial but globally non-trivial families can in principle not be classified by a representing object in the category of spaces that we are working with, as long as the Grothendieck topology which we use to say what a locally trivial family is is subcanonical:

for assume that a classifying object $K$ exists in our category and assume that at least one globally nontrivial but locally trivial family $\mathcal{E} \to X$ exists in the category.

Then by definition of local triviality there exists a covering sieve $\{U_i \to X\}$ such that the pullback of $\mathcal{E}$ to each $U_i$ is equivalent to the trivial family over $U_i$. Accordingly, it will be represented by the trivial element in $Hom(U_i, K)$. But if $\{U_i \to X\}$ is a covering sieve in a subcanonical topology, then it follows from the sheaf property of the representable presheaf $Hom(-,K)$ that there is a unique element in $Hom(X,K)$ whose restriction to all the $U_i$ represents the trivial family. But the trivial family over $X$ has this property and hence must be that unique element. Accordingly, it would follow that there is no non-trivial family classified by $K$ over $X$, which is a contradiction and shows that the “fine moduli space” $K$ cannot exist under the given assumptions.

Notice well the two assumptions that were made to make this argument work:

1. the topology with which we glue families must be subcanonical: this is the assumption that fails for the situations in which one would speak of classifying spaces instead of moduli spaces:

For instance vector spaces do certainly have nontrivial automorphisms. A (topological, say) family of vector space is a vector bundle. So a naive application of the above argument might lead one to conclude that there cannot be a classifying space of vector bundles!

But of course it is a standard fact that there there are topological spaces $\mathcal{B} U(n)$ such that homotopy classes of continuous maps $X \to \mathcal{B} U(n)$ classify isomorphism classes of rank-$n$ vector bundles. But this means that the functor that assigns families of vector bundles to topological space is – while not representable in Top – representable in the homotopy category $Ho(Top)$

$Rank n VectBund(X)/iso \simeq Hom_{Ho(Top)}(X, \mathcal{B} U(n)) \,.$

This might make the classifying space $\mathcal{B} U(n)$ look like something that deserves to be called a fine moduli space. But one should beware that the standard Grothendieck topology of Top, which is subcanonical in Top is not so in $Ho(Top)$: the functor $Hom_{Ho(Top)}(-, \mathcal{B} U(m))$ has a non-standard opinion about what it means to glue two topological spaces to a third one. That’s how in $Ho(Top)$ the above argument that “automorphisms prevent a classifying object” breaks down.

In fact, there is a theorem by Danny Stevenson and David Roberts, extending a theorem by John Baez and Danny Stevenson that shows that large classes of principal ∞-bundles, even, do have classifying topological spaces in this sense. These are objects that not only have automorphisms, but have automorphisms of automorphisms, etc, and hence seem to contradict the above slogan in a maximal way.

But these examples also indicate why it may be misguided to think of the “classifying spaces” such as $\mathcal{B} U(n)$ as being moduli spaces in the first place: the homotopy category $Ho(Top)$ in fact regards its objects – even though they are presented as topological spaces – not really as spaces but as the ∞-groupoids that they are equivalent to under the homotopy hypothesis. So in that sense objects such as $\mathcal{B} U(n)$ are not classifying spaces, but are classifying groupoids after all.

2. Another situation that makes the above argument break down is if the category of “spaces” that one works with is such that gluing trivial families by nontrivial automorphisms never produces globally nontrivial families. For instance because the category of “spaces” is such that no nontrivial families exist at all. This happens for instance in the case that we take a “space” to be simply a (finite, say) set. In the category Set of (finite) sets every fiber bundle $\mathcal{E} \to X$ (i.e. every surjection such that all fibers are isomorphic to a typical fiber $F$ ) are necessarily globally trivial in that they are isomorphic to $F \times X \to X$. And in fact, there is a “fine moduli space” in this context (inside the category of non-necessarily finite sets), namely the set $\mathbb{N}$ of natural numbers: the family $(\mathcal{E} \to X) \in Set$ with typical fiber the $n$-element set is classified by the map that sends all $x \in X$ to $n \in \mathbb{N}$.

## Examples

A bit of elementary exposition of these ideas is at