Английская Википедия: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 <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 p₂ 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 varieties or schemes.

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 category 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 quotients (e.g., <math>G/H</math>) and GIT quotients (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. Шаблон:MathSciNet Шаблон:ISBN

See also

• Quotient by an equivalence relation
• Quotient stack

Шаблон:Algebraic-geometry-stub