Categorical quotient

In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism $\pi :X\to Y$ that

(i) is invariant; i.e.,

\pi \circ \sigma =\pi \circ p_{2}

where

\sigma :G\times X\to X

is the given group action and p₂ is the projection.

(ii) satisfies the universal property: any morphism

X\to Z

satisfying (i) uniquely factors through

\pi

One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.

Note $\pi$ need not be surjective. Also, if it exists, a categorical quotient is unique up to a canonical isomorphism. In practice, one takes C to be the category of varieties or the category of schemes over a fixed scheme. A categorical quotient $\pi$ is a universal categorical quotient if it is stable under base change: for any $Y'\to Y$ , $\pi ':X'=X\times _{Y}Y'\to Y'$ is a categorical quotient.

A basic result is that geometric quotients (e.g., $G/H$ ) and GIT quotients (e.g., $X/\!/G$ ) are categorical quotients.

References

Mumford, David; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR 1304906 ISBN 3-540-56963-4

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