1 Category of Additive Fractions
Let us recall first the necessary concepts that enter in the definition of a category of additive
fractions.
1.1 Dense Subcategory
Definition 1.1. A full subcategory 𝒜 of an Abelian category 𝒞 is called dense if for any
exact sequence in 𝒞:
X is in 𝒜 if and only if both X′ and X′′ are in 𝒜.
1.2 Remark 0.1
One can readily prove that if X is an object of the dense subcategory 𝒜 of 𝒞 as defined above, then
any subobject XQ, or quotient object of X, is also in 𝒜.
1.3 System of morphisms ΣA
Let 𝒜 be a dense subcategory (as defined above) of a locally small Abelian category 𝒞, and let us
denote by ΣA (or simply only by Σ – when there is no possibility of confusion) the system of all
morphisms s of 𝒞 such that both kers and cokers are in 𝒜.
One can then prove that the category of additive fractions 𝒞Σ of 𝒞 relative to Σ exists.
1.4 Quotient Category
Definition 1.2. A quotient category of 𝒞 relative to 𝒜, denoted as 𝒞∕𝒜, is defined as the
category of additive fractions 𝒞Σ relative to a class of morphisms Σ := ΣA in 𝒞.
1.4.1 Remark 0.2
In view of the restriction to additive fractions in the above definition, it may be more appropriate
to call the above category 𝒞∕𝒜 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.
