Let $X$ be a stack of $n$groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be DeligneMumford, aside from having an etale atlas from an algebraic space, one must also impose certain separability conditions on its diagonal. In DGAV, Lurie remarks that what he defines as higher DeligneMumford stacks lack separation axioms, but that they may be added in by hand later. My question is, what separation axioms should be added? Here, I do not mean "separable" or "quasicompact". What I mean to ask is on which morphisms do I put the appropriate separability conditions? It seems to be that simply imposing them on the diagonal may be too naive, but perhaps I am wrong, hence this question.
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