Exactly how is 'the diagonal is representable' used for algebraic stacks... - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:40:12Z http://mathoverflow.net/feeds/question/57296 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57296/exactly-how-is-the-diagonal-is-representable-used-for-algebraic-stacks Exactly how is 'the diagonal is representable' used for algebraic stacks... David Roberts 2011-03-03T22:53:17Z 2011-03-18T15:22:13Z <p>...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for <strike>the</strike> <em>a</em> presenting algebraic groupoid $X_1 \rightrightarrows X_0$?</p> <p>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. (<b>Edit:</b> or can all (stable under pullback) properties of the diagonal be so described - and also used?)</p> <p>Pointers (in comments) to any relevant questions where example situations are discussed in detail would be appreciated.</p> <p>(NB This is a spin-off from comments at <a href="http://mathoverflow.net/questions/57057/what-about-stacks-of-categories-in-algebraic-geometry-ii" rel="nofollow">this question</a>.)</p> http://mathoverflow.net/questions/57296/exactly-how-is-the-diagonal-is-representable-used-for-algebraic-stacks/57362#57362 Answer by David Carchedi for Exactly how is 'the diagonal is representable' used for algebraic stacks... David Carchedi 2011-03-04T14:38:27Z 2011-03-04T14:38:27Z <p>If <code>$X_1 \rightrightarrows X_0$</code> is a groupoid and <code>$\mathcal{X}$</code> is the associated stack, consider the atlas <code>$p:X_0 \to \mathcal{X}$</code>. If you form the weak $2$-pullback of <code>$p \times p:X_0 \times X_0 \to \mathcal{X} \times \mathcal{X}$</code> against the diagonal <code>$\Delta:\mathcal{X} \to \mathcal{X} \times \mathcal{X}$</code> you get by first projection <code>$\left(s,t\right):X_1 \to X_0 \times X_0$</code>. But <code>$p \times p$</code> is an atlas for <code>$\mathcal{X} \times \mathcal{X}$</code> so it follows that <code>$\Delta$</code> is representable, and it has "property $P$" if and only if <code>$\Delta$</code> does.</p>