ag.algebraic geometry - Proper definition of a moduli problem

Since not many people have had anything to say, I thought I might make a few remarks. But beware that this is all what I've passively picked up over the years---it's not the result of an actual study of things.

I think the right definition of a moduli problem is a fibered category $p:Eto B$. This should be thought of as a family of categories parametrized by $B$, just like you think of a map $X to S$ of spaces as being a family of spaces parametrized by $S$. Here is an example: $E$ is the category whose objects are maps of schemes $Xto S$ making $X$ a family of elliptic curves over $S$ (i.e. an abelian scheme of relative dimension 1), and whose maps are the (hopefully) evident cartesian squares; $B$ is the category of schemes, and the functor $Eto B$ sends an object $Xto S$ to $S$. The fibered structure is given by pull back: given $Xto S$ in $E$ and a map $S'to S$ in $B$, we get the object $Xtimes_S S'to S'$ (which maps under $p$ to $S'$). If we let $E_S$ denote the fiber of this fibered category over an object $S$, then in this example, $E_S$ is the category of families of elliptic curves parametrized by $S$. Thus the fibered category encodes the data of all possible families of elliptic curves and how they behave under base change. So I hope my point that the fibered category is the moduli problem seems reasonable.

You then say the fibered category is representable if there is an object $U$ of $E$ such that for any $S$ in $B$, the pull back functor from the discrete category consisting of the set $mathrm{Hom}(S,p(U))$ to the category $E_S$ is an equivalence. This is pretty unlikely. For instance, it implies that each category $E_S$ is discrete---all maps are isomorphisms and all automorphisms are the identity. This is certainly not the case with the elliptic curve example. Every elliptic curve has a nontrivial automorphism given by the inverse map with respect to the group structure.

Another version of representability of a fibered category is the following. Let $F:Btomathrm{Sets}$ denote the functor which sends an object $S$ to the set of isomorphism classes of objects of $E_S$. Then the moduli problem is (weakly?) representable if the functor $F$ is representable. This definition is surely weaker in general than the one above, but it is often equivalent in the examples that people look at in algebraic geometry. For example, the two are probably equivalent if each $E_S$ is a discrete category.

If the moduli problem is not representable, then you get into other issues, such as whether you have effective descent with respect to some topology on $B$ (i.e. $E$ is a "sheaf of categories" over $B$), and if so, stack-theoretic issues, such as whether $E$ can be represented by a category object in $E$, and if so, whether it's a groupoid object.

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