In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory.
Mac Lane[1] says Alexander Grothendieck[2] defined the abelian category, but there is a reference[3] that says Eilenberg's disciple, Buchsbaum, proposed the concept in his PhD thesis,[4] and Grothendieck popularized it under the name "abelian category".
This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature.
The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between abelian categories.
In his Tōhoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category A might satisfy.
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.
Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category.
In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism.
For example, the poset of subobjects of any given object A is a bounded lattice.
The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A.
If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A.
in an abelian category, flatness refers to the idea that
Abelian categories are the most general setting for homological algebra.
called simple objects (meaning the only sub-objects of any
can be decomposed as a direct sum (denoting the coproduct of the abelian category)
Some abelian categories found in nature are semi-simple, such as
For example, the category of representations of the Lie group
In fact, this is true for any unipotent group[8]pg 112.
There are numerous types of (full, additive) subcategories of abelian categories that occur in nature, as well as some conflicting terminology.
Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor.
The essential image of I is a full, additive subcategory, but I is not exact.
In fact, much of category theory was developed as a language to study these similarities.