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 that

(i) is invariant; i.e., where is the given group action and p2 is the projection.
(ii) satisfies the universal property: any morphism satisfying (i) uniquely factors through .

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

Note 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 is a universal categorical quotient if it is stable under base change: for any , is a categorical quotient.

A basic result is that geometric quotients (e.g., ) and GIT quotients (e.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. MR1304906 ISBN 3-540-56963-4

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