Iasked me the question what the interpretation of the irreducibility of a moduli space is for the functor it represents. For proper, there is the valuative criterion and for (formally) smooth, there is the infinitesimal lifting property, but I don't know something similar for irreducible. Can someone shed some light on this?



functorial meaning of irreducibility of a moduli space
Your question is a bit vague, but let me try to say something. Being irreducible is a global property so there can be no local characterization like being smooth, regular, etc. If $\mathcal{M}$ is the "space" representing your functor, we say that it is irreducible if it admits a surjective map from an irreducible variety (this is just topological). 


If you think about what it means to be irreducible, we could also say that any two closed points of $\mathcal{M}$ can be connected by an irreducible variety. From the functorial point of view, this would mean that any two objects being parametrized live in a family over some irreducible variety. So for example, saying that $M\_g$ (the moduli space of smooth geneus $g$ curves is irreducible, is the same as saying that for any two smooth curves $C\_1, C\_2$ of genus $g$ (say over a field), there is a family $f: S \rightarrow B$ such that $B$ is irreducible, every geometric fiber is a smooth genus $g$ curve, and both $C\_1$ and $C\_2$ are fibers of $f$. In fact, you can take $B$ to be a curve.


