Abelian category

In the mathematical field of algebra and related fields is meant by an abelian category, a category that behaves in some key areas such as the category of abelian groups. To a lesser extent this is also true for additive categories.

Definition

It is a category together with the structure of an abelian group on each Morphismenmenge for objects.

Is a präadditive category if the following additional conditions are met:

  • The composition of morphisms is biadditiv, ie for morphisms and apply or, where the additions are indicated in the Morphismengruppen each with the same symbol .

Is an additive category if it is präadditiv and the following additional conditions are met:

  • There is a null object.
  • There are finite products.

Is an abelian category if it is präadditiv and in addition the following ( stronger ) conditions are met:

  • There is a null object.
  • There are (finite) Biprodukte, i.e. for any two objects there is an object together with morphisms and so
  • There are kernels and cokernels.
  • Every monomorphism is a kernel, every epimorphism is a cokernel.

Importance

Abelian categories are an important tool to generalize statements about abelian groups; for example, the Fünferlemma or Schlangenlemma valid in every abelian category. Abelian categories are also the natural context for homological algebra.

Properties

For abelian categories is:

  • The category is balanced: A morphism is an isomorphism if it is a monomorphism and an epimorphism, ie a Bimorphismus.
  • Every morphism has a substantially unique factorization into an epimorphism and a monomorphism.
  • The homomorphism and isomorphism theorems apply.

Examples

  • Each unitary ring is the Morphismenmenge präadditiven a category with a single object.

Additive is:

  • The category Div of divisible groups: The core of a homomorphism is always the null object ( with Nullhomomorphismus ), even if it is not injective. Therefore, the core is not a canonical projection, although it is on the other hand is a Monomorphismus.

Abelian include:

  • The category Ab of abelian groups.
  • The category of vector spaces for a body.
  • The category of -modules for a ring.
  • The category of sheaves of abelian groups on a topological space.
  • The category of finite abelian groups, the category of finitely generated abelian groups, more generally, the category of finitely generated modules over a Noetherian ring.

Embedding theorems

The close relationship to the abelian groups goes so far that one can perceive objects of an abelian category using a suitable functor as a special abelian groups ( embedding theorem of Mitchell):

  • For every small abelian category, there is a faithful functor exact.
  • For every small abelian category, there is a ring and a fully faithful exact functor from the category of -modules.

History

First approaches to the definition of the term " abelian category " are from S. Eilenberg and S. Mac Lane from the early 50s. However, the breakthrough came with A. Grothendieck 's epoch articles Sur quelques points d' algèbre homologique from the year 1957.

Product (category theory) Snake lemma Fundamental theorem on homomorphisms Mitchell's embedding theorem Samuel Eilenberg 1950s Alexander Grothendieck Peter J. Freyd
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