# Derived Functors of a Half-Exact Functor

This question is mainly posed out of sheer curiosity rather than any practical applications.

An additive functor F:C→DF:C\to D is called half-exact if for every short exact sequence
0→L→M→N→0
0\to L\to M\to N\to 0

There is an exact sequence
FL→FM→FN
FL\to FM\to FN

When FF is in fact left exact (resp. right exact), one can then define the right derived functors (resp. left derived functors) RiFR^i F (resp. LiFL_i F). These create long exact sequences which “fill in” the missing exactness in a “natural” way (i.e., in such a way that morphisms of short exact sequences induce morphisms of long exact sequences).

I am interested in the generalization to the case where FF is neither left exact nor right exact. Namely: do there exist derived functors RiFR^i F and LiFL_i F such that i) RiF=L−iFR^i F = L_{-i} F, and ii) for every short exact sequence of objects in CC there is an induced long exact sequence
…→L1FN→L→M→N→R1FL→…
…\to L_1 FN\to L\to M\to N\to R_1 FL\to …

which is natural in the sense mentioned above? And if so, how would one go about defining these functors?

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related question math.stackexchange.com/questions/1507311/half-exact-functor . Instead of “there is an exact sequence…” you should say “the sequence… is exact”. Obviously you want that the maps in second sequence are induced from those of the first one?
– syzygy
2 days ago

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