In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism π : X → Y that
(i) is invariant; i.e.,
π ∘ σ = π ∘ p 2 where
σ : G × X → X is the given group action and
p2 is the projection.(ii) satisfies the universal property: any morphism
X → Z satisfying (i) uniquely factors through
π .
One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.
Note π 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 π is a universal categorical quotient if it is stable under base change: for any Y ′ → Y , π ′ : X ′ = X × Y Y ′ → 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.
