View source for Quotient space of an algebraic stack

In algebraic geometry, the '''quotient space''' of an [[algebraic stack]] ''F'', denoted by |''F''|, is a [[topological space]] which as a set is the set of all integral substacks of ''F'' and which then is given a "[[Zariski topology]]": an open subset has a form <math>|U| \subset |F|</math> for some open substack ''U'' of ''F''.<!-- TODO: G ́erard Laumon and Laurent Moret-Bailly. Champs alg ́ebriques, uses a different but equivalent definition? --><ref>In other words, there is a natural bijection between the set of all open immersions to ''F'' and the set of all open subsets of <math>|F|</math>.</ref>

The construction <math>X \mapsto |X|</math> is functorial; i.e., each morphism <math>f: X \to Y</math> of algebraic stacks determines a continuous map <math>f: |X| \to |Y|</math>.

An algebraic stack ''X'' is '''punctual''' if <math>|X|</math> is a [[point topological space|point]].

When ''X'' is a moduli stack, the quotient space <math>|X|</math> is called the [[moduli space]] of ''X''. If <math>f: X \to Y</math> is a morphism of algebraic stacks that induces a [[homeomorphism]] <math>f: |X| \overset{\sim}\to |Y|</math>, then ''Y'' is called ''a'' coarse moduli stack of ''X''. ("The" coarse moduli requires a universality.)

== References ==
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*H. Gillet, [https://www.sciencedirect.com/science/article/pii/0022404984900367 Intersection theory on algebraic stacks and Q-varieties], J. Pure Appl. Algebra 34 (1984), 193–240, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983).

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[[Category:Algebraic geometry]]