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A category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106301.png" /> in which for any two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106303.png" /> an Abelian group structure is defined on the set of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106304.png" />, such that the composition of morphisms
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A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group structure is defined on the set of morphisms $\Hom_{\mathfrak C}(X,Y)$, such that the composition of morphisms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106305.png" /></td> </tr></table>
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$$\Hom_{\mathfrak C}(X,Y)\times\Hom_{\mathfrak C}(Y,Z)\to\Hom_{\mathfrak C}(X,Z)$$
  
is a bilinear mapping. Another necessary condition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106306.png" /> includes a null object (zero object, cf. [[Null object of a category|Null object of a category]]) as well as the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106307.png" /> of any two objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a0106309.png" />.
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is a bilinear mapping. Another necessary condition is that $\mathfrak C$ includes a null object (zero object, cf. [[Null object of a category|Null object of a category]]) as well as the product $X\times Y$ of any two objects $X$ and $Y$.
  
In an additive category the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063010.png" /> of any two objects exists. It is isomorphic to their product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063011.png" />. The [[Dual category|dual category]] to an additive category is also an additive category.
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In an additive category the direct sum $X\oplus Y$ of any two objects exists. It is isomorphic to their product $X\times Y$. The [[Dual category|dual category]] to an additive category is also an additive category.
  
A functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063012.png" /> from an additive category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063013.png" /> into an additive category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063014.png" /> is said to be additive if, for any objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063017.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063018.png" /> is a homomorphism of Abelian groups. An additive category is said to be pre-Abelian if a kernel (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]) and a [[Cokernel|cokernel]] exist for any morphism.
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A functor $F\colon\mathfrak C\to\mathfrak C_1$ from an additive category $\mathfrak C$ into an additive category $\mathfrak C_1$ is said to be additive if, for any objects $X$ and $Y$ in $\mathfrak C$, the mapping $F\colon\Hom_{\mathfrak C}(X,Y)\to\Hom_{\mathfrak C_1}(F(X),F(Y))$ is a homomorphism of Abelian groups. An additive category is said to be pre-Abelian if a kernel (cf. [[Kernel of a morphism in a category|Kernel of a morphism in a category]]) and a [[Cokernel|cokernel]] exist for any morphism.
  
If for a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063019.png" /> in an additive category there exists an image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063020.png" /> and a co-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063021.png" />, then there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063022.png" /> such that the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063023.png" /> splits as the composition
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If for a morphism $u\colon X\to Y$ in an additive category there exists an image $\operatorname{Im}(u)$ and a co-image $\operatorname{Coim}(u)$, then there exists a unique morphism $u\colon\operatorname{Coim}(u)\to\operatorname{Im}(u)$ such that the morphism $u$ splits as the composition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063024.png" /></td> </tr></table>
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$$X\to\operatorname{Coim}(u)\to\operatorname{Im}(u)\to Y.$$
  
An Abelian category is additive by definition. Examples of non-Abelian additive categories are the category of topological modules over a given topological ring with respect to the morphisms which are continuous linear mappings, and also the category of Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063025.png" /> with a filtration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063026.png" /> with respect to the morphisms which are group homomorphisms preserving the filtration.
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An Abelian category is additive by definition. Examples of non-Abelian additive categories are the category of topological modules over a given topological ring with respect to the morphisms which are continuous linear mappings, and also the category of Abelian groups $\Gamma$ with a filtration $\Gamma=\Gamma_0\supset\Gamma_1\supset\dots\supset\Gamma_n=\{0\}$ with respect to the morphisms which are group homomorphisms preserving the filtration.
  
 
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====Comments====
 
====Comments====
The requirement, in the definition of an additive category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063027.png" /> above, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010630/a01063028.png" /> possesses a null object as well as the product of any two objects in it, is not standard. Most authors drop this requirement, and take an additive category to mean a category satisfying only the requirement in the first sentence of the main text above.
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The requirement, in the definition of an additive category $\mathfrak C$ above, that $\mathfrak C$ possesses a null object as well as the product of any two objects in it, is not standard. Most authors drop this requirement, and take an additive category to mean a category satisfying only the requirement in the first sentence of the main text above.

Latest revision as of 23:53, 10 December 2018

A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group structure is defined on the set of morphisms $\Hom_{\mathfrak C}(X,Y)$, such that the composition of morphisms

$$\Hom_{\mathfrak C}(X,Y)\times\Hom_{\mathfrak C}(Y,Z)\to\Hom_{\mathfrak C}(X,Z)$$

is a bilinear mapping. Another necessary condition is that $\mathfrak C$ includes a null object (zero object, cf. Null object of a category) as well as the product $X\times Y$ of any two objects $X$ and $Y$.

In an additive category the direct sum $X\oplus Y$ of any two objects exists. It is isomorphic to their product $X\times Y$. The dual category to an additive category is also an additive category.

A functor $F\colon\mathfrak C\to\mathfrak C_1$ from an additive category $\mathfrak C$ into an additive category $\mathfrak C_1$ is said to be additive if, for any objects $X$ and $Y$ in $\mathfrak C$, the mapping $F\colon\Hom_{\mathfrak C}(X,Y)\to\Hom_{\mathfrak C_1}(F(X),F(Y))$ is a homomorphism of Abelian groups. An additive category is said to be pre-Abelian if a kernel (cf. Kernel of a morphism in a category) and a cokernel exist for any morphism.

If for a morphism $u\colon X\to Y$ in an additive category there exists an image $\operatorname{Im}(u)$ and a co-image $\operatorname{Coim}(u)$, then there exists a unique morphism $u\colon\operatorname{Coim}(u)\to\operatorname{Im}(u)$ such that the morphism $u$ splits as the composition

$$X\to\operatorname{Coim}(u)\to\operatorname{Im}(u)\to Y.$$

An Abelian category is additive by definition. Examples of non-Abelian additive categories are the category of topological modules over a given topological ring with respect to the morphisms which are continuous linear mappings, and also the category of Abelian groups $\Gamma$ with a filtration $\Gamma=\Gamma_0\supset\Gamma_1\supset\dots\supset\Gamma_n=\{0\}$ with respect to the morphisms which are group homomorphisms preserving the filtration.

References

[1] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[2] A. Grothendieck, "Sur quelques points d'algèbrique homologique" Tohôku Math. J. , 9 (1957) pp. 119–221
[3] L. Gruson, "Complétion abélienne" Bull. Sci. Math. (2) , 90 (1966) pp. 17–40


Comments

The requirement, in the definition of an additive category $\mathfrak C$ above, that $\mathfrak C$ possesses a null object as well as the product of any two objects in it, is not standard. Most authors drop this requirement, and take an additive category to mean a category satisfying only the requirement in the first sentence of the main text above.

How to Cite This Entry:
Additive category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_category&oldid=19009
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article