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# additive functor

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 $A,B$ in $\mathcal{A}$, the function

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

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 $A$ and codomain $B$, then

$F(f+g)=F(f)+F(g).$ |

For example, the hom functor $\hom(A,-)$ where $A$ 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…

Related:

PreAdditiveFunctors, CategoryOfAdditiveFractions

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## Mathematics Subject Classification

18E05*no label found*

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