An (additive) functor F from an additive category A to an additive category B
is said to be objective, provided any morphism f in A with F(f) = 0 factors
through an object K with F(K) = 0. In this paper 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, in particular we
want do 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_2F_1 with F_1 a full and dense triangle functor and F_2 a faithful triangle
functor. It turns our that the behaviour of splitting monomorphisms (and
splitting epimorphisms) plays a decisive role
