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. A basic result is that geometric quotients (e.g., ) and GIT quotients (e.g., ) are categorical quotients.