0.1 Essential data: Dense subcategory
Definition 0.1. 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.
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.
0.1.1 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.
Definition 0.2. The 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.
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 Remark 0.1 is also applicable in the context of the more
general definition of a quotient category.
