category of additive fractions
Let us recall first the necessary concepts
that enter in the definition
of a category of additive fractions.
Definition 1.1
A full subcategory

of an
abelian category

is called
dense if for any exact sequence in

:

is in

if and only if both

and

are in

.
One can readily prove that if
is an object of the dense subcategory
of
as defined above, then any subobject
, or quotient object of
, is also in
.
Let
be a dense subcategory (as defined above) of a locally small Abelian category
,
and let us denote by
(or simply only by
- when there is no possibility of confusion) the system of all morphisms
of
such that both
and
are in
.
One can then prove that the category of additive fractions
of
relative to
exists.
Definition 1.2
A
quotient category of
relative to
, denoted as

, is defined as the category of additive fractions

relative to a class of morphisms

in

.
In view of the restriction to additive fractions in the above definition, it may be more appropriate to call the above category
an additive quotient category.
This would be important in order to avoid confusion with the more general notion of
quotient category
-which is defined as a category of fractions. Note however that the above remark is also applicable in the context of the more general definition of a quotient category.
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As of this snapshot date, this entry was owned by bci1.