Categorical quotient

In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group 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 p₂ 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 was the construction of a categorical quotient for varieties or schemes.

Note $\pi$ 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 $\pi$ is a universal categorical quotient if it is stable under base change: for any $Y^{\prime }\to Y$, ${\pi }^{\prime }:X^{\prime }=X{\times }_Y{Y}^{\prime }\to Y^{\prime }$ is a categorical quotient.

A basic result is that geometric quotients (e.g., $G/H$) and GIT quotients (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. Template:MathSciNet Template:ISBN

See also

• Quotient by an equivalence relation
• Quotient stack

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