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 formula_1 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 formula_5 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 formula_5 is a universal categorical quotient if it is stable under base change: for any formula_8, formula_9 is a categorical quotient.
A basic result is that geometric quotients (e.g., formula_10) and GIT quotients (e.g., formula_11) are categorical quotients.