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

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

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.

is a universal categorical quotient if it is stable under base change: for any

A basic result is that geometric quotients (e.g.,

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