In mathematics, the quotient of an abelian category A by a Serre subcategory B is the category whose objects are those of A and whose morphisms from X to Y are given by the direct limit
lim
→
H
o
m
A
(
X
′
,
Y
/
Y
′
)
over subobjects
X
′
⊆
X
and
Y
′
⊆
Y
such that
X
/
X
′
,
Y
′
∈
B
. The quotient A/B will then be an Abelian category, and there is a canonical functor
Q
:
A
→
A
/
B
sending an object X to itself and a morphism
f
:
X
→
Y
to the corresponding element of the direct limit with X'=X and Y'=0. This Abelian quotient satisfies the universal property that if C is any other Abelian category, and
F
:
A
→
C
is an exact functor such that F(b) is a zero object of C for each
b
∈
B
, then there is a unique exact functor
F
¯
:
A
/
B
→
C
such that
F
=
F
¯
∘
Q
.
