In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism that
- (i) is invariant; i.e., where is the given group action and p2 is the projection.
- (ii) satisfies the universal property: any morphism satisfying (i) uniquely factors through .
One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.
Note 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 is a universal categorical quotient if it is stable under base change: for any , is a categorical quotient.
A basic result is that geometric quotients (e.g., ) and GIT quotients (e.g., ) are categorical quotients.