Objective triangle functors

Abstract

An additive functor $$F:\mathcal{A} \to \mathcal{B}$$ between additive categories is said to be objective, provided any morphism f in $$\mathcal{A}$$ with F(f) = 0 factors through an object K with F(K) = 0. We concentrate on triangle functors between triangulated categories. The first aim of this paper is to characterize objective triangle functors F in several ways. Second, we are interested in the corresponding Verdier quotient functors $$V_F :\mathcal{A} \to \mathcal{A}/KerF$$ , in particular we want to know under what conditions V F is full. The third question to be considered concerns the possibility to factorize a given triangle functor F = F 2 F 1 with F 1 a full and dense triangle functor and F 2 a faithful triangle functor. It turns out that the behavior of splitting monomorphisms and splitting epimorphisms plays a decisive role.