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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 Deligne-Mumford, aside from having an etale atlas from an algebraic space, one must also impose certain separation conditions on its diagonal. In DGA-V, Lurie remarks that what he defines as higher Deligne-Mumford stacks lacks 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 "separated" or "quasicompact". What I mean to ask is on which morphisms do I put the appropriate separation 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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If you look at the last lemma here [a link to the stacks project] and the comment preceding it, you can see that the most appropriate morphisms to impose conditions on are probably the higher diagonals.


The lemma gives a bunch of conditions for a morphism of algebraic stacks to have various properties. The conditions on the diagonal and second diagonal of the morphism. Precisely, if these two diagonals are universally closed, then the morphism is separated; if they are quasi-compact, then the morphism is quasi-separated; if they are unramified, then the morphism is DM (``Deligne--Mumford''); if they are monomorphisms, then the morphism if representable by algebraic spaces.

(The point is that for a morphism of algebraic stacks, the second diagonal is always a monomorphism, and hence the diagonals beyond the second are always isomorpisms, and hence satisfy every condition you might impose. But for higher stacks, the second diagonal is presumably not always a monomorphism, and hence a natural thing to do would be to consider morphisms all of whose higher diagonals are universally closed, or quasi-compact, or unramified, or monomorphisms.)

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