Abelian category

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 . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \operatorname {Hom} _{\mathcal {C}}(X,Y)}$ ${\displaystyle X,Y\in \operatorname {Ob} {\mathcal {C}}}$

${\displaystyle {\mathcal {C}}}$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 . ${\displaystyle f,f_{1},f_{2}\in \operatorname {Hom} _{\mathcal {C}}(X,Y)}$ ${\displaystyle g,g_{1},g_{2}\in \operatorname {Hom} _{\mathcal {C}}(Y,Z)}$ ${\displaystyle g\circ (f_{1}+f_{2})=(g\circ f_{1})+(g\circ f_{2})}$ ${\displaystyle (g_{1}+g_{2})\circ f=(g_{1}\circ f)+(g_{2}\circ f)}$ ${\displaystyle +}$

${\displaystyle {\mathcal {C}}}$is an additive category if it is pre-additive and the following conditions are also met:

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

${\displaystyle {\mathcal {C}}}$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 ${\displaystyle X_{1},X_{2}}$ ${\displaystyle X_{1}\oplus X_{2}}$ ${\displaystyle p_{\nu }\colon X_{1}\oplus X_{2}\to X_{\nu }}$ ${\displaystyle i_{\nu }\colon X_{\nu }\to X_{1}\oplus X_{2}}$ ${\displaystyle \nu =1,2}$

${\displaystyle p_{\nu }\circ i_{\nu }=\operatorname {id} _{X_{\nu }}}$ and ${\displaystyle i_{1}\circ p_{1}+i_{2}\circ p_{2}=\operatorname {id} _{X_{1}\oplus X_{2}}}$

applies and that with forms a product and with a co- product . ${\displaystyle X_{1}\oplus X_{2}}$ ${\displaystyle p_{\nu }}$ ${\displaystyle i_{\nu }}$

• There are cores and coke cores .
• Every monomorphism is a nucleus, every epimorphism a coke nucleus .

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 . ${\displaystyle i\circ p}$ ${\displaystyle p}$ ${\displaystyle i}$
• 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. ${\displaystyle f\colon \mathbb {Q} /\mathbb {Z} \to G}$ ${\displaystyle f}$ ${\displaystyle \pi \colon \mathbb {Q} \to \mathbb {Q} /\mathbb {Z} }$

Abelsch are for example:

• The category from the Abelian groups .
• The category of - vector spaces for a body . ${\displaystyle K}$ ${\displaystyle K}$
• The category of - modules for a ring . ${\displaystyle A}$ ${\displaystyle A}$
• 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 . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}\to \mathbf {Ab} }$
• For every small Abelian category there is a ring and a fully faithful exact functor of in the category of modules. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle A}$

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

• Peter Freyd : Abelian Categories. An Introduction to the Theory of Functors. Harper & Row, New York NY et al. 1964.
• Alexander Grothendieck : Sur quelques points d'algèbre homologique. In: Tohôku Mathematical Journal. Ser. 2, Vol. 9, No. 2, 1957, pp. 119-221, doi : 10.2748 / tmj / 1178244839 .
• Saunders Mac Lane : Categories for the Working Mathematician (= Graduate Texts in Mathematics. 5). Springer, New York NY et al. 1971, ISBN 0-387-90036-5 .