Suppose that $X$ is a separated Deligne-Mumford stack, say over a base scheme.
Is there some standard notation for the geometric quotient of $X$? I've tried using $[X]$ but have had complaints.
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$\begingroup$ Do you mean some sort of course space, good moduli space (in the sense of Alper), or something like that? One standard notation is to denote the stack by $\mathscr X$ or $\mathcal X$ and the scheme by $X$. $\endgroup$– R. van Dobben de BruynCommented Dec 14, 2019 at 0:41
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$\begingroup$ Yes, that is a standard convention, and it works if there is only one stack (``the stack'') in the picture. But it is inconvenient in general. For example, what if I have stacks $\mathcal X$ and $\mathcal Y$, and need to consider the stack $Mor({\mathcal X},{\mathcal Y})$ of morphisms? Or $Bun_G^s$ is often used for a stack of stable $G$-bundles over some variety; must I denote it by $\mathcal{X}$ before taking its geometric quotient? $\endgroup$– inkspotCommented Dec 14, 2019 at 7:57
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$\begingroup$ You could use $X^c$ maybe? $\endgroup$– Ariyan JavanpeykarCommented Dec 14, 2019 at 9:22
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$\begingroup$ How does that work if $X=Bun_G^s$? Do I write $Bun_G^{sc}$? In this context of a group the $sc$ suggests ``simply connected''. Is $(Bun_G^s)^c$ better than $[Bun_G^s]$? And in any case my question was whether there is some standard (that is, already in existence) notation. $\endgroup$– inkspotCommented Dec 14, 2019 at 9:58
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$\begingroup$ I think $X^c$ or $X^{coarse}$ is quite common. Also, $(Bun^s_G)^s$ seems fine to me. The notation $[X]$ might seem a bit too close to $|X|$ which I think usually denotes the set of points of the stack $X$; see stacks.math.columbia.edu/tag/04XE $\endgroup$– Ariyan JavanpeykarCommented Dec 14, 2019 at 13:52
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