On a Strange Homotopy Category

Abstract

Let \( \mathcal{C} \) be an additive category in which each morphism has a kernel. It is proved that the homotopy category of the category of complexes over \( \mathcal{C} \) which are concentrated in degrees 2, 1, 0 and are exact in degrees 2 and 1 is Abelian. Under assumption that a category \( \mathcal{C} \) is Abelian, this result was obtained earlier by considering the heart of a suitable t-structure on the homotopy category of \( \mathcal{C} \).