In Deligne and Mumford's famous 1969 paper, The irreducibility of the space of curves of given genus, definition 4.6 (that of algebraic stacks) has the following footnote:
This definition is the "right" one only for quasi-separated stacks. It will however be sufficient for our purposes.
Note that their definition of an algebraic stack is:
- A stack on $Sch$ with the etale topology such that
- The diagonal is representable, and
- there is a representable etale surjection from a scheme.
and it immediately follows the definition of what it means for a stack to be quasi-separated, but they clearly do not mention quasi-separability here.
What do they mean by the footnote? What is the "right" definition for stacks which are not quasi-separated?