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Do general fixes and cleanup. - using AWB (9774)

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In [[algebraic geometry]], given a [[category (mathematics)]] ''C'', a '''categorical quotient''' of an object ''X'' with action of a [[group (mathematics)|group]] ''G'' is a [[morphism]] <math>\pi: X \to Y</math> that
:(i) is invariant; i.e., <math>\pi \circ \sigma = \pi \circ p_2 </math> where <math>\sigma: G \times X \to X</math> is the given group action and ''p''<sub>2</sub> is the projection.
:(ii) satisfies the universal property: any morphism <math>X \to Z</math> satisfying (i) uniquely factors through <math>\pi</math>.
One of the main motivations for the development of [[geometric invariant theory]] was the construction of a categorical quotient for [[algebraic variety|varieties]] or [[scheme (mathematics)|scheme]]s.

Note <math>\pi</math> 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 cateogy of schemes over a fixed scheme. A categorical quotient <math>\pi</math> is a '''universal categorical quotient''' if it is stable under base change: for any <math>Y' \to Y</math>, <math>\pi': X' = X \times_Y Y' \to Y'</math> is a categorical quotient.

A basic result is that [[geometric quotient]]s (e.g., <math>G/H</math>) and [[GIT quotient]]s (e.g., <math>X/\!/G</math>) 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. {{MathSciNet|id=1304906}} ISBN 3-540-56963-4

[[Category:Algebraic geometry]]

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