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In algebraic geometry, given a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
''C'', a categorical quotient of an object ''X'' with
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''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 ''p''2 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 In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
was the construction of a categorical quotient for
varieties Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
or schemes. Note \pi need not be surjective. Also, if it exists, a categorical quotient is unique up to a canonical
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. 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 quotient In algebraic geometry, a geometric quotient of an algebraic variety ''X'' with the action of an algebraic group ''G'' is a morphism of varieties \pi: X \to Y such that :(i) For each ''y'' in ''Y'', the fiber \pi^(y) is an orbit of ''G''. :(ii) The ...
s (e.g., G/H) and
GIT quotient In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of ...
s (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. {{ISBN, 3-540-56963-4


See also

*
Quotient by an equivalence relation In mathematics, given a category ''C'', a quotient of an object ''X'' by an equivalence relation f: R \to X \times X is a coequalizer for the pair of maps :R \ \overset\ X \times X \ \overset\ X,\ \ i = 1,2, where ''R'' is an object in ''C'' and ...
*
Quotient stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. Th ...
Algebraic geometry