As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:

### Deformation theory

"Moduli of Curves" defines a versal deformation space as a deformation $\phi: X \rightarrow Y$ such that for any other deformation $\xi: X \rightarrow Z$ and for every point in $Z$ there exists an open set (in the complex topology) $U$ such that the pullback of $\phi$ via $f: U \rightarrow Y$ is $\xi$ restricted to $U$. (I imagine that in general instead of an open set one takes open etale covers - is this true?)

### Moduli Spaces

In moduli spaces, versality has always meant a space such that instead of the geometric points being in 1-1 correspondence with the objects we're interested in (over the field of the geo. point), each object is going have several geometric points in the versal space corresponding to it.

### Question

Are these two notions related? If so - how?

Higher Topos Theory, which gives us not only categories and higher categories cofibered in groupoids, but also recovers deformation spaces and moduli spaces using $\infty$-groupoids. $\endgroup$ – Harry Gindi Mar 8 '10 at 20:57