Let $F: Ab \rightarrow Ab$ be an additive functor and $I$ a finite index set, show that $F(\sum_{i \in I} A_i) \approx \sum_{i \in I} F(A_i)$

Question

I'm self learning Algebraic Topology from Rotman's Algebraic Topology and I've come across this problem:

Let $F: Ab \rightarrow Ab$ be an additive functor and $I$ a finite index set, show that $F(\sum_{i \in I} A_i) \approx \sum_{i \in I} F(A_i)$

I can't think of a way to start this. Additive functors are functors such that if $f,g: B \rightarrow B'$ then $F(f + g) = F(f) + F(g)$. I can't think of a way to show that there's an isomorphism relationship here.

Anyone have any ideas? Thorough explanations would be helpful as I'm self learning this.

Answer 1

I think you will be satisfied with this link to Overflow : Additive covariant functors preserve direct sums.

If you're interested in modules, there is proof adapted to categories of modules in "Rings and Categories of Modules" of F.W. Anderson and K.R. Fuller p.180. You will also find a proof for infinite index with the $Hom$ functors (which is an additive functor).