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.

## 1 Answer

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.)