In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism
π
:
X
→
Y
that
(i) is invariant; i.e.,
π
∘
σ
=
π
∘
p
2
where
σ
:
G
×
X
→
X
is the given group action and
p2 is the projection.
(ii) satisfies the universal property: any morphism
X
→
Z
satisfying (i) uniquely factors through
π
.
One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.
Note
π
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
π
is a universal categorical quotient if it is stable under base change: for any
Y
′
→
Y
,
π
′
:
X
′
=
X
×
Y
Y
′
→
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.
