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 $$\pi: X \to Y$$ that
 * (i) is invariant; i.e., $$\pi \circ \sigma = \pi \circ p_2 $$ where $$\sigma: G \times X \to X$$ is the given group action and p2 is the projection.
 * (ii) satisfies the universal property: any morphism $$X \to Z$$ satisfying (i) uniquely factors through $$\pi$$.

One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.

Note $$\pi$$ 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 $$\pi$$ is a universal categorical quotient if it is stable under base change: for any $$Y' \to Y$$, $$\pi': X' = X \times_Y Y' \to Y'$$ 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.