An additive functor F:A→B between additive categories is said to be objective,provided any morphism f in A with F(f)=0 factors through an object K with F(K)=0.We concentrate on triangle functors between triangulated ...An additive functor F:A→B between additive categories is said to be objective,provided any morphism f in 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 VF:A→A/Ker F,in particular we want to know under what conditions VF is full.The third question to be considered concerns the possibility to factorize a given triangle functor F=F2F1with F1a full and dense triangle functor and F2a faithful triangle functor.It turns out that the behavior of splitting monomorphisms and splitting epimorphisms plays a decisive role.展开更多
An additive functor F： A→B between additive categories is objective if any morphism f in A with F（f） = 0 factors through an object K with F（K） = 0. We consider when a triangle functor in an adjoint pair is objec...An additive functor F： A→B between additive categories is objective if any morphism f in A with F（f） = 0 factors through an object K with F（K） = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint （whatever left adjoint or right adjoint） is full or dense. We Mso give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary a^tjoint pair （F, G） between categories which are not necessarily additive, we give a sufficient and necessary condition such that F （resp., G） is full or faithful.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11271251 and 11431010)Specialized Research Fund for the Doctoral Program of Higher Education(GrantNo.20120073110058)
文摘An additive functor F:A→B between additive categories is said to be objective,provided any morphism f in 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 VF:A→A/Ker F,in particular we want to know under what conditions VF is full.The third question to be considered concerns the possibility to factorize a given triangle functor F=F2F1with F1a full and dense triangle functor and F2a faithful triangle functor.It turns out that the behavior of splitting monomorphisms and splitting epimorphisms plays a decisive role.
文摘An additive functor F： A→B between additive categories is objective if any morphism f in A with F（f） = 0 factors through an object K with F（K） = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint （whatever left adjoint or right adjoint） is full or dense. We Mso give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary a^tjoint pair （F, G） between categories which are not necessarily additive, we give a sufficient and necessary condition such that F （resp., G） is full or faithful.
