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Commutative Diagram Structures And Satisfaction Axiom Group Category Definition Identity Element Function Class Abelian Group Addition Basic Polynomial Types Of Morphisms Categorical Direct Product Universal Property Zero Object Additive Category

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Abelian Category

An abelian category is a category $\mathcal{A}$ satisfying the following axioms. Because the later axioms rely on terms whose definitions involve the earlier axioms, we will intersperse the statements of the axioms with such auxiliary definitions as needed.

Axiom 1. For any two objects in $\mathcal{A}$ , the set of morphisms $\operatorname{Hom}(A,B)$ admits an abelian group structure, with group operation denoted by , satisfying the following naturality requirement: given any diagram of morphisms

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar[r]^{f} & B \ar@/_/[r]_{g_2} \ar@/^/[r]^{g_1} & C \ar[r]^h & D } } \end{xy}$

we have

and

. That is, composition of morphisms must distribute over addition in $\operatorname{Hom}(\cdot,\cdot)$ .

The identity element in the group $\operatorname{Hom}(\cdot,\cdot)$ will be denoted by 0.

Axiom 2. $\mathcal{A}$ has a zero object.

Axiom 3. For any two objects in $\mathcal{A}$ , the categorical direct product $A \times B$ exists in $\mathcal{A}$ .

Given a morphism $f\colon A \to B$ in $\mathcal{A}$ , a kernel of is a morphism $i\colon X \to A$ such that:

For any other morphism $j\colon X' \to A$ such that , there exists a unique morphism $j'\colon X' \to X$ such that the diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & X' \ar[d]^j \ar@{-->}[dl]_{j'} \\ X \ar[r]^i & A \ar[r]^f & B } } \end{xy}$
commutes.

Likewise, a cokernel of

is a morphism $p\colon B \to Y$ such that:

For any other morphism $j\colon B \to Y'$ such that , there exists a unique morphism $j'\colon Y \to Y'$ such that the diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar[r]^f & B \ar[r]^p \ar[d]_j & Y \ar@{-->}[dl]^{j'} \\ & Y' } } \end{xy}$
commutes.

Axiom 4. Every morphism in $\mathcal{A}$ has a kernel and a cokernel.

The kernel and cokernel of a morphism in $\mathcal{A}$ will be denoted $\ker(f)$ and $\operatorname{cok}(f)$ , respectively. (Some texts use the notation $\operatorname{coker}(f)$ for cokernel.) By the universal properties above, the kernel and cokernel of are only unique up to isomorphism, but by abuse of notation we write $\ker(f)$ for a representative element of this isomorphism class.

A morphism $f\colon A \to B$ in $\mathcal{A}$ is called a monomorphism if, for every morphism $g\colon X \to A$ such that , we have . Similarly, the morphism is called an epimorphism if, for every morphism $h\colon B \to Y$ such that , we have .

Axiom 5. $\ker(\operatorname{cok}(f)) = f$ for every monomorphism in $\mathcal{A}$ .

Axiom 6. $\operatorname{cok}(\ker(f)) = f$ for every epimorphism in $\mathcal{A}$ .

Remark. Equivalently, an abelian category is an additive category such that Axioms 4-6 are satisfied.

1	Category
1	Commutative Diagram
1	Basic Polynomial
1	Definition
1	Structures And Satisfaction
1	Class
1	Function
1	Axiom
1	Addition
1	Identity Element
1	Abelian Group
1	Group
2	Types Of Morphisms
2	Categorical Direct Product
3	Universal Property
4	Zero Object
9	Additive Category