ag.algebraic geometry - Stacks and sheaves

Let me see if I understand your example correctly: you are fixing $X$ and $Y$, families
of curves over $S$, and now you are considering the functor which maps an $S$-scheme $T$
to the set of $T$-isomorphisms $f^*X to f^*Y$ (where $f$ is the map from $T$ to $S$).

If I have things straight, then this functor shouldn't be so bad to think about, because it is actually representable, by an Isom scheme. In other words, there is an $S$-scheme
$Isom_S(X,Y)$ whose $T$-valued points, for any $f:T to S$, are precisely the $T$-isomorphisms
from $f^*X$ to $f^*Y$. (One can construct the Isom scheme by looking inside a
certain well-chosen Hilbert scheme.)

One way to think about this geometrically is as follows: one can imagine that two
curves over $k$ (a field) are isomorphic precisely when certain invariants coincide
(e.g. for elliptic curves, the $j$-invariant). (Of course this is a simplification,
and the whole point of the theory of moduli spaces/schemes/stacks is to make it precise,
but it is a helpful intuition.) Now if we have a family $X$ over $S$, these invariants
vary over $S$ to give a collections of functions on $S$ (e.g. a function $j$ in the
genus $1$ case), and similarly with $Y$. Now $X$ and $Y$ will have isomorphic
fibres precisely at those points where the invariants coincide, so if we look
at the subscheme $Z$ of $S$ defined by the coincidence of the invariants,
we expect that $f^*X$ and $f^*Y$ will be isomorphic precisely if the map $f$
factors through $Z$. Thus $Z$ is a rough approximation to the Isom scheme.

It is not precisely the Isom scheme, because curves sometimes have non-trivial
automorphisms, and so even if we know that $X_s$ and $Y_s$ are isomorphic for
some $s in S$, they may be isomorphic in more than one way. So actually the
Isom scheme will be some kind of (possibly ramified) finite cover of $Z$.

Of course, if one pursues this line of intuition much more seriously, one will
recover the notions of moduli stack, coarse moduli space, and so on.

Added: The following additional remark might help:

The families $X$ and $Y$ over $S$ correspond to a map $phi:S to {mathcal M}_g times {mathcal M}_g$. The stack which maps a $T$-scheme to $Isom_T(f^*X, f^*Y)$ can then seen to be the fibre product of the map $phi$ and the diagonal
$Delta:{mathcal M}_g to {mathcal M}_g times {mathcal M}_g$.

In the particular case of ${mathcal M}_g$ the fact that this fibre product is representable is part of the condition that ${mathcal M}_g$ be an algebraic stack.

But in general, the construction you describe is the construction of a fibre product
with the diagonal. This might help with the geometric picture, and make the relationship to Mike's answer clearer. (For the latter:note that the path space into $X$ has a natural
projection to $Xtimes X$ (take the two endpoints), and the loop space is the fibre product
of the path space with the diagonal $Xto Xtimes X$.)

• Next evolution - Did we first have swimming birds or flying birds?
• Previous set theory - Does every non-empty set admit a group structure (in ZF)?