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The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
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Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups.
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Consider the category Ab of abelian groups and group homomorphisms.
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In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: where the rows are exact sequences and 0 is the zero object.
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More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category.
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Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors LnT are defined.
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With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups.
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Maybe the most important injective module is the abelian group Q/Z. It is an injective cogenerator in the category of abelian groups, which means that it is injective and any other module is contained in a suitably large product of copies of Q/Z. So in particular, every abelian group is a subgroup of an injective one.
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If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category.
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Homomorphism groups: To every pair A, B of abelian groups one can assign the abelian group Hom(A,B) consisting of all group homomorphisms from A to B. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor Abop × Ab → Ab (where Ab denotes the category of abelian groups with group homomorphisms).
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Consider the category D of homomorphisms of abelian groups.
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The constant presheaf with value A is the presheaf that assigns to each non-empty open subset of X the value A, and all of whose restriction maps are the identity map A → A. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A. In certain cases, the set A may be replaced with an object A in some category C (e.g. when C is the category of abelian groups, or commutative rings).
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There was perhaps a more direct route available: the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules.
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If X is a topological space, we can consider the abelian category of all sheaves of abelian groups on X. The functor which associates to each sheaf F the group of global sections F(X) is left-exact.
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Every continuous map f : X → Y between topological spaces induces a functor f ∗ from the category of sheaves (of sets, or abelian groups, or rings...) on X to the corresponding category of sheaves on Y, the direct image functor.
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This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.
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Birkhoff's representation theorem relating distributive lattices and partial orders Pontryagin duality gives a duality on the category of locally compact abelian groups: given any such group G, the character group χ(G) = Hom (G, S1) given by continuous group homomorphisms from G to the circle group S1 can be endowed with the compact-open topology.
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To prove this, one uses the peculiar properties of the abelian group Q/Z to construct an injective cogenerator in the category of left R-modules.
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For the category of groups, just turn all additive notation below into multiplicative notation, and note that commutativity of abelian group is never used.
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The five lemma is not only valid for abelian categories but also works in the category of groups, for example.
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The motivating prototype example of an abelian category is the category of abelian groups, Ab.
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In Algebra, specifically within the category of abelian groups, there exist the concepts of torsion theory and radical.
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Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms
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Note that the enriched structure on hom-sets is a consequence of the three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature.
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Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group.
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