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 [math]\displaystyle{ \pi: X \to Y }[/math] that

(i) is invariant; i.e., [math]\displaystyle{ \pi \circ \sigma = \pi \circ p_2 }[/math] where [math]\displaystyle{ \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]\displaystyle{ X \to Z }[/math] satisfying (i) uniquely factors through [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \pi }[/math] is a universal categorical quotient if it is stable under base change: for any [math]\displaystyle{ Y' \to Y }[/math], [math]\displaystyle{ \pi': X' = X \times_Y Y' \to Y' }[/math] is a categorical quotient.

A basic result is that geometric quotients (e.g., [math]\displaystyle{ G/H }[/math]) and GIT quotients (e.g., [math]\displaystyle{ 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. MR1304906 ISBN:3-540-56963-4

See also

• Quotient by an equivalence relation
• Quotient stack