PlanetMath: additive functor

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additive functor

(Definition)

Let $\mathcal{A}$ and $\mathcal{B}$ be ab-categories. A functor $F:\mathcal{A}\to \mathcal{B}$ is called an additive functor if, for any objects in $\mathcal{A}$ , the function

$\displaystyle F_{(A,B)}: \hom(A,B)\to \hom(F(A),F(B))$

given by $F_{(A,B)}(f)=F(f)$ is a group homomorphism. In other words, if $f,g: A\to B$ are two morphisms with common domain

and codomain

, then

$\displaystyle F(f+g)=F(f)+F(g).$

For example, the hom functor $\hom(A,-)$ where is an object in an abelian category, is additive.

Remark. It can be shown that any exact functor between abelian categories is additive.

More to come...

"additive functor" is owned by CWoo.

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See Also: preadditive functor, category of additive fractions

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Cross-references: exact functor, additive, abelian category, hom functor, codomain, domain, morphisms, group homomorphism, function, objects, functor, ab-categories
There are 4 references to this entry.

This is version 2 of additive functor, born on 2008-06-12, modified 2008-06-13.
Object id is 10698, canonical name is AdditiveFunctor.
Accessed 384 times total.

Classification:

AMS MSC:

18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories)

Pending Errata and Addenda

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