Is there a good notion of `Separated Stack'?

Question

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, doesn't seem to provide an answer.

In a previous MO question 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. 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.

Answer 1

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).

Answer 2

One can first define a 'proper algebraic space' $X,$ using its 'underlying space' $|X|,$ and then define a morphism of algebraic spaces $f: X \to Y$ to be proper if for any affine (or just quasi-compact) scheme mapping into $Y,$ the fiber product gives a proper algebraic space. Finally define an Artin stack $X$ to be separated if the diagonal (which is representable) is proper, i.e. for any algebraic space mapping into $X \times X,$ the fiber product....

One can also define a 'proper Artin stack' similarly. See Laumon and Moret-Bailly.

Answer 3

Maybe I'm missing something, but I'm pretty sure "separated" for morphisms of schemes IS local on the target:

http://stacks.math.columbia.edu/tag/02KU

The problem with your reasoning is that if $U\hookrightarrow Y$ is the inclusion of an affine open and $f:X\rightarrow Y$ is your morphism, then the pullback of $f$ by $U$ may not be affine, and hence you can't conclude that $f|_U$ is separated.

Actually, it seems to me that the real obstacle to defining separated algebraic stacks by piggy-backing on the scheme version of separatedness is that for an algebraic stack $\mathcal{X}$ over a scheme $S$, the structure morphism $\mathcal{X}\rightarrow S$ is never representable unless $\mathcal{X}$ was itself already a scheme.

Answer 4

The work of Street "Two-dimensional sheaf theory" (JPAA 23 1982 251-270) is relevant I believe.

He has notions of 1-separated and 2-separated, and a 2-analog of the $\_^+$ functor which he calls $L$, with natural transformation $l:\text{identity}\to L$, such that

• $P$ is 1-separated iff $l_P:P\to LP$ is mono;
• for every $P$, $LP$ is 1-separated;
• $P$ is 2-separated iff $l_P:P\to LP$ is $\textit{chronic}$;
• for every 1-separated $P$, $LP$ is 2-separated;
• $P$ is a stack iff $l_P:P\to LP$ is an isomorphism;
• for every 2-separated $P$, $LP$ is a stack.

Here a morphism $f:P\to Q$ is called chronic when the functor $\hom(X,f):\hom(X,P)\to\hom(X,Q)$ is full, faithful and injective on objects for every $X$.

Comparing this with the properties of $\_^+$ seemingly leads to the conclusion that separatedness naturally splits into two notions, something like "$2/3$-separatedness" and "$4/3$-separatedness".

See also the (later) Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first? here on MO.

Update

After comments by David Roberts I must confess I no longer believe it is relevant. However I still don't understand why it is irrelevant, so let me leave it for a while. Sorry.