Categorical quotient

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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.


  • 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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