Moduli Spaces, Representable Functors, and Fibered Products

The idea of a moduli space is that its points correspond to a geometric object of interest. The underlying set of the space is thus determined, but not its geometric structure. In most cases, one can describe a geometric structure in an ad hoc manner, but it is not immediately obvious how to rigorously formulate the idea that the resulting object is “the” moduli space for the problem. This is the problem addressed by the theory of representable functors, with the key idea being that we specify not only the objects of interest, but also what families of these objects should look like.