A scheme is *separated* if the diagonal inclusion $X \to X \times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks?


My usual stack reference, Anton Gerashchenko's [stack notes](http://stacky.net/files/written/Stacks/Stacks.pdf), doesn't seem to provide an answer. 


In a [previous MO question](https://mathoverflow.net/questions/9043/why-is-this-not-an-algebraic-space) several related notions came up. The most similar is quasi-separated where you require the diagonal to be quasi-compact. You can check wikipedia for some relevant [algebraic geometry terminology](http://en.wikipedia.org/wiki/Glossary_of_scheme_theory). How does this compare to separatedness?


The main obstacle that I can see in defining separated for stacks is that the property of a map of schemes $X \to Y$ being separated does not appear to be local in the target. Since maps between affines are separated, it seems that every map of schemes is *locally* separated. This means that we shouldn't expect the usual trick of replacing an algebraic stack by a scheme which covers it to work very well. 



Is there a good notion of `Separated Stack'?
Look at Def. 4.7 of Deligne--Mumford for the definition when $X$ is DM: they define $X$
to be separated if $X \to X \times X$ is proper (or equivalently, finite).


