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 ${\displaystyle \pi :X\to Y}$ 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 ${\displaystyle \pi }$ 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 ${\displaystyle \pi }$ is a universal categorical quotient if it is stable under base change: for any ${\displaystyle Y'\to Y}$, ${\displaystyle \pi ':X'=X\times _{Y}Y'\to Y'}$ is a categorical quotient.

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