Abelian category

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In the mathematical sub-area of algebra and related areas, an Abelian category is understood to be a category that behaves like the category of Abelian groups in some essential aspects . To a lesser extent, this also applies to additive categories .

definition

Let it be a category together with the structure of an Abelian group on each set of morphisms for objects .

is a pre-additive category if the following conditions are also met:

  • The composition of morphisms is bi-additive, that is to say for morphisms and applies or , the additions in the morphism groups being denoted by the same symbol .

is an additive category if it is pre-additive and the following conditions are also met:

is an Abelian category if it is pre-additive and the following (stronger) conditions are also met:

  • There is a zero object.
  • There are (finite) biproducts , i. H. for every two objects there is an object together with morphisms and for , so that
and
applies and that with forms a product and with a co- product .

meaning

Abelian categories are an important tool for generalizing statements about Abelian groups; for example, the five lemma or the snake lemma apply in every Abelian category. Abelian categories are also the natural context for homological algebra .

properties

The following applies to Abelian categories:

  • The category is balanced : a morphism is an isomorphism if and only if it is a monomorphism and an epimorphism, i.e. a bimorphism.
  • Each morphism has an essentially unique factorization into an epimorphism and a monomorphism .
  • The homomorphism and isomorphism theorems apply.

Examples

  • Each unitary ring is the set of morphisms of a pre-additive category with a single object.

Additive is:

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

Abelsch are for example:

  • The category from the Abelian groups .
  • The category of - vector spaces for a body .
  • The category of - modules for a ring .
  • The category of sheaves of Abelian groups in 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 sets

The close relationship to the Abelian groups goes so far that objects of an Abelian category can be understood as special Abelian groups with the help of a suitable functor ( Mitchell's embedding theorem ):

  • For every little Abelian category there is an exact faithful functor .
  • For every small Abelian category there is a ring and a fully faithful exact functor of in the category of modules.

history

The first approaches to defining the term "Abelian category" come from S. Eilenberg and S. Mac Lane in the early 1950s . The breakthrough came with A. Grothendieck's epoch-making article Sur quelques points d'algèbre homologique from 1957.

literature