Objective triangle functors

TL;DR

This paper investigates the properties of objective triangle functors between triangulated categories, providing characterizations, conditions for fullness of Verdier quotients, and factorization criteria involving full, dense, and faithful functors.

Contribution

It offers new characterizations of objective triangle functors, analyzes when Verdier quotient functors are full, and explores factorization conditions based on splitting morphisms.

Findings

Characterizations of objective triangle functors

Conditions for Verdier quotient functors to be full

Factorization criteria involving full, dense, and faithful functors

Abstract

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.