Quotient space of an algebraic stack

From HandWiki

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 |U||F| for some open substack U of F.[1] The construction X|X| is functorial; i.e., each morphism f:XY of algebraic stacks determines a continuous map f:|X||Y|.

An algebraic stack X is punctual if |X| is a point.

When X is a moduli stack, the quotient space |X| is called the moduli space of X. If f:XY is a morphism of algebraic stacks that induces a homeomorphism f:|X||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 |F|.