The category $K(\mathcal{C})$

DEFINITION 11.3   [1, 5.1] For any additive category $\mathcal{C}$ , we define $K(\mathcal{C})$ to be

1. $\operatorname{Ob}(K(\mathcal{C}))=\operatorname{Ob}(C(\mathcal{C}))$ (that means, objects of $K(\mathcal{C})$ are complexes).
2. For any objects $X^\bullet, Y^\bullet$ of $K(\mathcal{C})$ , we define
   $$\operatorname{Hom}_{K(\mathcal{C})} (X^\bullet,Y^\bullet) = \operatorname{Hom}_{C(\mathcal{C})} (X^\bullet,Y^\bullet)/$$Homotopy

Even if $\mathcal{C}$ is abelian, $K(\mathcal{C})$ is no longer abelian in general [1, 5.7]. But $K(\mathcal{C})$ has distinguished triangles, which are triangles isomorphic to standard triangles.