Abelian category

A category that displays some of the characteristic properties of the category of all Abelian groups. Abelian categories were introduced as the basis for an abstract construction of homological algebra [4]. A category $ \mathfrak A $ is said to be Abelian [2] if it satisfies the following axioms:

A0. A null object exists (cf. Null object of a category).

A1. Each morphism has a kernel (cf. Kernel of a morphism in a category) and a cokernel.

A2. Each monomorphism is a normal monomorphism, so that it occurs as the kernel of a morphism; each epimorphism is a normal epimorphism.

A3. A product and a coproduct exist for each pair of objects (cf. Product of a family of objects in a category).

In defining an Abelian category it is often assumed that $ \mathfrak A $ is a locally small category. The coproduct of two objects $ A $ and $ B $ of an Abelian category is also know as the direct sum of these objects and is denoted by $ A \oplus B $, $ A \amalg B $ or $ A \dot{+} B $.

Examples of Abelian categories.

1) The dual category of an Abelian category is also an Abelian category.

2) The category $ {} _ {R} \mathfrak M $ of all unitary left modules over an arbitrary associative ring $ R $ with a unit element and all $ R $- module homomorphisms is an Abelian category (e.g. the category of all Abelian groups).

3) Any full subcategory of an Abelian category which contains for each one of its morphisms also the kernel and cokernel of that morphism, and which contains for each pair of objects $ A $ and $ B $ also their product and coproduct, is an Abelian category.

The small Abelian categories are exhausted by the subcategories of the above type of categories $ {} _ {R} \mathfrak M $ of unitary left modules. Accordingly, the following Mitchell theorem is valid: For each small Abelian category there exists a full exact imbedding into some category $ {} _ {R} \mathfrak M $.

4) Any category of diagrams $ \mathfrak F ( \mathfrak D, \mathfrak A ) $, with diagram scheme $ \mathfrak D $ over an Abelian category $ \mathfrak A $, is an Abelian category. In the scheme $ \mathfrak D $ one may distinguish the set $ C $ of commutativity relations, i.e. the set of pairs $ ( \phi , \psi ) $ of paths $ \phi = ( \phi _ {1} \dots \phi _ {n} ) $, $ \psi = ( \psi _ {1} \dots \psi _ {m} ) $ in $ \mathfrak D $ with a common begin and end. Then the complete subcategory of the category $ \mathfrak F ( \mathfrak D , \mathfrak A ) $ generated by all those diagrams $ D: \mathfrak D \rightarrow \mathfrak A $ that satisfy

$$ D ( \phi ) = D ( \phi _ {1} ) \dots D ( \phi _ {n} ) = \ D ( \psi _ {1} ) \dots D ( \psi _ {m} ) = D ( \psi ) $$

is an Abelian category. In particular, if $ \mathfrak D $ is a small category and if the set $ C $ consists of all pairs of the form $ ( \alpha \beta , \gamma ) $ with $ \gamma = \alpha \beta $, then the corresponding subcategory is the Abelian category of one-place covariant functors from $ \mathfrak D $ into $ \mathfrak A $( cf. Functor).

Suppose that a null object exists in a small category $ \mathfrak D $. A functor $ F : \mathfrak D \rightarrow \mathfrak A $ will then be called normalized if it takes a null object into a null object. The complete subcategory of the category of functors generated by the normalized functors will then be an Abelian category. In particular, if $ \mathfrak D $ is a category the objects of which are all integers plus the null object $ N $, while the non-null non-identity morphisms form a sequence

$$ {} \dots \rightarrow ( -1 ) \mathop \rightarrow \limits ^ { {d _ {-1} }} 0 \mathop \rightarrow \limits ^ { {d _ {0} }} 1 \mathop \rightarrow \limits ^ { {d _ {1} }} 2 \rightarrow \dots $$

in which $ d _ {n} d _ {n+1} = 0, n = 0 , \pm 1 , \pm 2 \dots $ then the corresponding subcategory generated by the normalized functors is called the category of complexes over $ \mathfrak A $. On the category of complexes there are defined the additive functors $ Z ^ {n} , B ^ {n} , H ^ {n} $ of $ n $- dimensional cycles, $ n $- dimensional boundaries and $ n $- dimensional homologies, respectively, with values in $ \mathfrak A $. These constitute the basis for the development of homological algebra.

5) A full subcategory $ \mathfrak A _ {1} $ of an Abelian category $ \mathfrak A $ is said to be dense if it contains all subobjects (cf. Subobject) and quotient objects (cf. Quotient object) of its objects, and if for the exact sequence

$$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $$

$ B \in \mathop{\rm Ob} \mathfrak A _ {1} $ if and only if $ A, C \in \mathop{\rm Ob} \mathfrak A _ {1} $. The quotient category $ \mathfrak A / \mathfrak A _ {1} $ is then constructed as follows. Let $ ( R, \mu ] $ be a subobject of the direct sum $ A \oplus B $ with projections $ \pi _ {1} , \pi _ {2} $ and let the square

$$ \begin{array}{rcl} R & \mathop \rightarrow \limits ^ { {\mu \pi _ {1} }} & A \\ \mu \pi _ {2} \downarrow &{} &\downarrow \alpha \\ B & \mathop \rightarrow \limits _ \beta & X \\ \end{array} $$

be co-universal (i.e. a co-fibred product). The subobject $ (R, \mu ] $ is called an $ \mathfrak A _ {1} $- subobject if $ \mathop{\rm Coker} \mu \pi _ {1} $, $ \mathop{\rm Ker} \beta \in \mathfrak A _ {1} $. Two $ \mathfrak A _ {1} $- subobjects are equivalent if they contain some $ \mathfrak A _ {1} $- subobject. By definition, the set $ H _ {\mathfrak A / \mathfrak A _ {1} } (A, B) $ consists of the equivalence classes of $ \mathfrak A _ {1} $- subobjects. The usual multiplication of binary relations is compatible with the equivalence relation thus introduced, which makes it possible to construct the quotient category $ \mathfrak A / \mathfrak A _ {1} $, which is an Abelian category. The exact functor $ T: \mathfrak A \rightarrow \mathfrak A / \mathfrak A _ {1} $ is defined by assigning to each morphism $ \alpha : A \rightarrow B $ its graph in $ A \oplus B $. A subcategory $ \mathfrak A _ {1} $ is said to be a localizing subcategory if $ T $ has a complete univalent right-adjoint functor $ Q : \mathfrak A / \mathfrak A _ {1} \rightarrow \mathfrak A $.

6) For any topological space $ X $ the category of left $ G $- modules over $ X $, where $ G $ is a sheaf of rings with unit element over $ X $, is an Abelian category.

It is possible to introduce into any Abelian category $ \mathfrak A $ a partial sum of morphisms so that $ A $ becomes an additive category. For this reason the product and the coproduct of any pair of objects in an Abelian category are identical. Moreover, in defining an Abelian category it suffices to assume the existence of either products or coproducts. Any Abelian category is a bicategory with a unique bicategorical structure. These properties characterize an Abelian category: A category $ \mathfrak A $ with finite products is Abelian if and only if it is additive and if any morphism $ \alpha $ has a kernel and a cokernel and can be decomposed into a product

$$ \alpha = \mathop{\rm Coker} ( \mathop{\rm Ker} \alpha ) \theta \mathop{\rm ker} ( \mathop{\rm Coker} \alpha ), $$

in which $ \theta $ is an isomorphism.

The Mitchell theorem quoted above constitutes the underlying principle of the so-called "diagram-chasing" method in an Abelian category: Any proposition about commutative diagrams that is valid for all categories of left modules $ {} _ {R} \mathfrak M $ and that is a consequence of the exactness of certain sequences of morphisms, is valid in all Abelian categories.

In a locally small Abelian category, the $ \mathfrak A $- subobjects of an arbitrary object form a Dedekind lattice. If products (or coproducts) of any family of objects exist in $ \mathfrak A $, this lattice will be complete. These conditions are known to be met if there is a generating object $ U $ in $ \mathfrak A $ and if the coproducts

$$ \amalg _ {i \in I } U _ {i} ,\ U _ {i} = U, $$

exist for any set $ I $. These conditions are satisfied, for instance, by Grothendieck categories (cf. Grothendieck category), which are equivalent to the quotient categories of the category of modules by a localizing subcategory (the Gabriel–Popescu theorem).

References

[1]	I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[2]	P. Freyd, "Abelian categories: An introduction to the theory of functors" , Harper & Row (1964)
[3]	P. Gabriel, "Des categories Abéliennes" Bull. Soc. Math. France , 90 (1962) pp. 323–448
[4]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohôku Math. J. , 9 (1957) pp. 119–221

Comments

Composition of morphisms is written from left to right in this article; i.e. $ \phi \psi $ denotes the composition of $ \phi : A \rightarrow B $, $ \psi : B \rightarrow C $. A dense subcategory is more often called a Serre subcategory.

References

[a1]	B. Mitchell, "Theory of categories" , Acad. Press (1965)
[a2]	N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)