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.