TITLE: Quotient maps in algebraic geometry
QUESTION [6 upvotes]: In topology, a quotient map is a surjective map $\pi:X\to Y$ such that $V\subseteq Y$ is open in $Y$ if and only if $\pi^{-1}(V)$ is open in X. This definition has the following nice property: If $\rho:X\to Z$ is a continuous map and $f:Y\to Z$ is any map of sets such that $\rho=f\circ \pi$, then $f$ is continuous. 
What is the corresponding concept of quotient map in the category of quasi-projective varieties? Is it only $\textit{surjective regular map}$?
A more concrete question: let $\pi:X\to Y$ be a surjective regular map of quasi-projective varieties. Is it true that for any regular map $\rho:X\to Z$, a map of sets $f:Y\to Z$ such that $\rho=f\circ\pi$ is necessarily a regular map?

REPLY [5 votes]: The answer to the concrete question is no. For example, take $X=\mathbb{P}^1=Z$ with $\rho$ the identity. Let $Y$ be a projective singular rational curve with a cusp and let $\pi:X\to Y$ be the normalization map. Then, $\pi$ is a bijection and thus we get a map of sets $f:Y\to Z$ with $\rho=f\circ\pi$, but $f$ is not regular.