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In algebraic geometry, given a category *C*, a **categorical quotient** of an object *X* with action of a group *G* is a morphism $$

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
*p*_{2}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.

Note 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 is a **universal categorical quotient** if it is stable under base change: for any , is a categorical quotient.

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

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