Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits an distinguished triangle $$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[1]$$ in the unbounded derived category $D(\mathcal{A})$ for some $B^{\bullet}\in D^-(\mathcal{A})$ and $C^{\bullet}\in D^+(\mathcal{A})$.

Can we do the same thing in the unbounded homotopy category $K(\mathcal{A})$?

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle $$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[1]$$ in the unbounded derived category $D(\mathcal{A})$ for some $B^{\bullet}\in D^-(\mathcal{A})$ and $C^{\bullet}\in D^+(\mathcal{A})$.

In other words, any complex can be decomposed into a complex bounded above and a complex bounded below.

Can we do the same thing in the unbounded homotopy category $K(\mathcal{A})$?

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits an distinguished triangle $$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[1]$$ in the unbounded derived category $D(\mathcal{A})$ for some $B^{\bullet}\in D^+(\mathcal{A})$ and $C^{\bullet}\in D^-(\mathcal{A})$.

Can we do the same thing in the unbounded homotopy category $K(\mathcal{A})$?

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits an distinguished triangle $$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[1]$$ in the unbounded derived category $D(\mathcal{A})$ for some $B^{\bullet}\in D^-(\mathcal{A})$ and $C^{\bullet}\in D^+(\mathcal{A})$.

Can we do the same thing in the unbounded homotopy category $K(\mathcal{A})$?