# 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 ${\displaystyle \pi :X\to Y}$ that

(i) is invariant; i.e., ${\displaystyle \pi \circ \sigma =\pi \circ p_{2}}$ where ${\displaystyle \sigma :G\times X\to X}$ is the given group action and p2 is the projection.
(ii) satisfies the universal property: any morphism ${\displaystyle X\to Z}$ satisfying (i) uniquely factors through ${\displaystyle \pi }$.

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

Note ${\displaystyle \pi }$ 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 ${\displaystyle \pi }$ is a universal categorical quotient if it is stable under base change: for any ${\displaystyle Y'\to Y}$, ${\displaystyle \pi ':X'=X\times _{Y}Y'\to Y'}$ is a categorical quotient.

A basic result is that geometric quotients (e.g., ${\displaystyle G/H}$) and GIT quotients (e.g., ${\displaystyle X/\!/G}$) 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. MR1304906 ISBN 3-540-56963-4