Categorical quotient explained

\pi: X \to Y that

(i) is invariant; i.e.,

\pi \circ \sigma = \pi \circ p _{2} where \sigma: G x 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' \to Y , \pi': X' = X x _{Y} Y' \to Y' 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.

See also

• Quotient by an equivalence relation
• Quotient stack