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 that
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 and GIT quotients are categorical quotients.
