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 ${\displaystyle |U|\subset |F|}$ for some open substack U of F.[1]

The construction ${\displaystyle X\mapsto |X|}$ is functorial; i.e., each morphism ${\displaystyle f:X\to Y}$ of algebraic stacks determines a continuous map ${\displaystyle f:|X|\to |Y|}$.

An algebraic stack X is punctual if ${\displaystyle |X|}$ is a point.

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

References

1. 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 ${\displaystyle |F|}$.

• H. Gillet, 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).