# Exactly how is 'the diagonal is representable' used for algebraic stacks...

...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for the a presenting algebraic groupoid $X_1 \rightrightarrows X_0$?

Once we know that given a stack $\mathcal{X}$ we have a smooth representable $X_0 \to \mathcal{X}$ where $X_0$ is a scheme, then we can talk about the algebraic groupoid $X_1 :=X_0\times_\mathcal{X} X_0 \rightrightarrows X_0$, which has source and target smooth maps. We thus have the map $(s,t)$, and can talk about its properties, such as being separated or whatever. We can specify its properties (such as having 'property P') by demanding that the diagonal $\Delta:\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable and has 'property P'. But where else is representability and property $P$ of $\Delta$ used, in a way that couldn't be otherwise derived from property $P$ of $(s,t)$? 'Everybody knows' that algebraic stacks and algebraic groupoids form the objects of two equivalent bicategories (there is a 1996 article by Dorette Pronk that makes this precise, and recent work by myself - available on the arXiv if you care - expands hers to be applied in more general situations). Thus I wonder what properties of the diagonal $\Delta$ are used that couldn't be instead derived from $(s,t)$ of a presenting groupoid. (Edit: or can all (stable under pullback) properties of the diagonal be so described - and also used?)

Pointers (in comments) to any relevant questions where example situations are discussed in detail would be appreciated.

(NB This is a spin-off from comments at this question.)

• I've always thought that the condition that the diagonal is representable is useful because it is equivalent to requiring that any map from a scheme to your stack is representable. In particular, it implies that the map from an atlas of the stack to the stack is representable, which is certainly important. Mar 4, 2011 at 2:14
• @Mike - but we can require representability of the atlas map separately, which is essential, and if pressed, require that any map from a scheme to the stack is representable. If it is this latter property which is used, then that is the sort of answer I am looking for. And any other reasons too, of course. Mar 4, 2011 at 3:00
• Since you can reconstruct the diagonal up to 2-isomorphism from a presenting groupoid, it seems tautological that there isn't any (isomorphism-invariant) property of the diagonal that can't be derived from properties of a presenting groupoid. Is that really the question you mean to ask? Also, I protest at your use of the word "the" in front of "presenting". Mar 4, 2011 at 5:41
• @Scott - regarding 'the' - whoops! I didn't mean that. And yes, I think you've hit the nail on the head: I really want to know if one can avoid using representability of the diagonal explicitly in favour of properties arising from a presenting groupoid. Mar 4, 2011 at 5:54
• One point (which doesn't answer your question, I know) is that representability of the diagonal follows from the existence of an atlas (i.e. a smooth surjective and representable morphism from an algebraic space $X\to \mathcal{X}$). See for example prop (4.3.2) of Champs Algébriques. Mar 4, 2011 at 11:54

If $X_1 \rightrightarrows X_0$ is a groupoid and $\mathcal{X}$ is the associated stack, consider the atlas $p:X_0 \to \mathcal{X}$. If you form the weak $2$-pullback of $p \times p:X_0 \times X_0 \to \mathcal{X} \times \mathcal{X}$ against the diagonal $\Delta:\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ you get by first projection $\left(s,t\right):X_1 \to X_0 \times X_0$. But $p \times p$ is an atlas for $\mathcal{X} \times \mathcal{X}$ so it follows that $\Delta$ is representable, and it has "property $P$" if and only if $\Delta$ does.
• What kind of property $P$ do you have in mind exactly? Maybe then I can give a better answer. May 24, 2011 at 11:36