Non-additive derived functors

Let F C E be a functor from a category C to a homological (Borceux-Bourn) or semi-abelian (Janelidze-M\'arki-Tholen) category E. We investigate conditions under which the homology of an object X in C with coefficients in the functor F, defined via projective resolutions in C, remains independent of the chosen resolution. Consequently, the left derived functors of F can be constructed analogously to the classical abelian case. Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn-Janelidze, originally introduced in the context of subtractive categories. This method is applicable when C is a pointed regular category with finite coproducts and enough projectives, provided these projectives are closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor F meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts - conditions that amount to additivity when C and E are abelian categories. Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.