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 $\backslash pi:\; X\; \backslash to\; Y$ that

:(i) is invariant; i.e., $\backslash pi\; \backslash circ\; \backslash sigma\; =\; \backslash pi\; \backslash circ\; p\_2$ where $\backslash sigma:\; G\; \backslash times\; X\; \backslash to\; X$ is the given group action and p₂ is the projection.

:(ii) satisfies the universal property: any morphism $X\; \backslash to\; Z$ satisfying (i) uniquely factors through $\backslash pi$.

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

Note $\backslash pi$ 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 $\backslash pi$ is a universal categorical quotient if it is stable under base change: for any $Y\text{'}\; \backslash to\; Y$, $\backslash pi\text{'}:\; X\text{'}\; =\; X\; \backslash times\_Y\; Y\text{'}\; \backslash to\; Y\text{'}$ is a categorical quotient.

A basic result is that geometric quotients (e.g., $G/H$) and GIT quotients (e.g., $X/\backslash !/G$) are categorical quotients.