I've been reading Mumford's Paper 'Picard Groups of Moduli Problems'. Stated in modern language, the most famous result from the paper is that the moduli stack of elliptic curves has Picard group $\mathbb Z/12$.
In the original paper, however, Mumford introduces various (Grothendieck pre)topologies, including one called the 'moduli topology' for the moduli problem of curves of genus $g$, defined on the category of all families of genus $g$ curves; i.e., on the stack $\mathcal M_g$ itself rather than on the base category.
It is clear that, when he wrote the paper, Mumford considered these topologies to be very important objects of study. But I'm having trouble relating something like the moduli topology to the modern idea of a stack.
How do the topologies in this paper relate to stacks? And can anyone point me towards an exposition of Mumford's argument in the language of stacks?