Triangulated Structures Induced by Triangle Functors

Given a triangle functor F : A → B , the authors introduce the half image hIm F , which is an additive category closely related to F . If F is full or faithful, then hIm F admits a natural triangulated structure. However, in general, one can not expect that hIm F has a natural triangulated structure. The aim of this paper is to prove that hIm F admits a natural triangulated structure if and only if F satisfies the condition (SM). If this is the case, hIm F is triangle-equivalent to the Verdier quotient A /Ker F .