Lemma 13.11.1. Let \mathcal{A} be an abelian category. The functor
H^0 : K(\mathcal{A}) \longrightarrow \mathcal{A}
is homological.
Proof. Because H^0 is a functor, and by our definition of distinguished triangles it suffices to prove that given a termwise split short exact sequence of complexes 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 the sequence H^0(A^\bullet ) \to H^0(B^\bullet ) \to H^0(C^\bullet ) is exact. This follows from Homology, Lemma 12.13.12. \square
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