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 In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

\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 or

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

s. Note

\pi

need not be

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...

. 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 quotients (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 ...

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

References

See also