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Consider an additive category $\mathcal{C}$. It is known that the category $Ch(\mathcal{C})$ of chain complexes in $\mathcal{C}$ is again an additive category and hence one can consider the category $Ch(Ch(\mathcal{C}))$. This category has many uses and its structure is well-understood since it is basically the category of double complexes.

Now let $K(\mathcal{C})$ be the homotopy category of chain complexes in $\mathcal{C}$, i.e. the category whose objects are chain complexes in $\mathcal{C}$ and whose morphisms are chain homotopy classes of morphisms in $Ch(\mathcal{C})$. The homotopy category is also additive, but it is moreover triangulated. As such its structure is also quite well-understood, and at the very least it has important applications in regards to the derived category $D(\mathcal{C})$, when $\mathcal{C}$ is abelian. My question is now the following:

What is known about $K(K(\mathcal{C}))$?

Specifically I would like to know whether there has been any research or published literature into the structure and properties of $K(K(\mathcal{C}))$ (or higher iterations), and whether such repeated homotopy categories have had any applications. I'll also gladly accept answers in which the results or applications require $\mathcal{C}$ to fulfill additional properties, like being abelian for example.

As an example, if I were asking about $Ch(Ch(\mathcal{C}))$ an appropriate answer would be to point me towards literature on homological algebra which treats double complexes and their basic properties like the salamander lemma, or more advanced topics like the spectral sequence of a double complex and its applications.

I should also add that I'll accept a negative answer as well, if that is the situation. A simple "Nope, nobody cares about those and nobody ever really used them explicitly" would suffice.

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I do not think that considering complexes over $K(C)$ is a good idea; I never met any mention of those. – Mikhail Bondarko Apr 23 '13 at 17:37

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