Additive functors of abelian groups induce functor.

Question

The additive functor of abelian cateogries $F:A\rightarrow B$ gives rise to a functor $Ch_{\geq 0}(F):Ch_{\geq 0}(A)\rightarrow Ch_{\geq 0}(B)$

I've come across this statement in much of my reading, but haven't been able to find an actual proof for it anywhere. Does anyone know where I can find a proof or can they give me a proof?

Answer 1

I don't know the details of the full story, but here is what I think the sketch is

Any additive functor $F : A \to B$ automatically induces a functor $F_* : \operatorname{Ch}_{\geq 0}(A)\rightarrow \operatorname{Ch}_{\geq 0}(B)$ simply by applying it to each object and arrow. Furthermore, this functor preserves chain homotopies.

If the abelian category is good enough, you can (?) construct

• A subcategory $C \subseteq \operatorname{Ch}_{\geq 0}(A)$ on which "quasi-isomorphism" is the same thing as "chain homotopy equivalence"
• A functor $Q : \operatorname{Ch}_{\geq 0}(A) \to C$
• A natural transformation $q : Q \to 1$ whose components are all quasi-isomorphisms

The point being that the homotopical functor you are searching for can be given simply by $F_*$ when restricted to $C$, and the deformation $(Q,q)$ lets us replace any chain complex with a quasi-isomorphic one from $C$.

Then, up to weak equivalence, the quasi-isomorphism preserving functor you're looking for is $F_*Q $

I think the story here is that you pick $C$ to be complexes of projectives and $Q$ to be a functor that picks out such a complex quasi-isomorphic to its input. Unfortunately, I don't recall the fine detail.