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If $V$ is a monoidal model category, then its homotopy category $\mathrm{Ho}(V)$ is a monoidal category. Similarly, if $M$ is a $V$-model category, then $\mathrm{Ho}(M)$ is a $\mathrm{Ho}(V)$-category.

On the other hand, if $M$ is a stable model category, then $\mathrm{Ho}(M)$ is a triangulated category. In "The additivity of traces in triangulated categories", May wrote down a list of compatibility axioms between a monoidal structure and a triangulation, and showed that if $V$ is a monoidal stable model category then $\mathrm{Ho}(V)$ satisfies these axioms.

One could easily write down versions of (most of) May's axioms relative to an enrichment rather than a monoidal structure, replacing the monoidal tensor product by the tensor of an enrichment, and the internal-homs with the cotensor and enriched-homs. Essentially the same proof would then show that if $V$ is a stable monoidal model category and $M$ is a (stable) $V$-model category, then the enrichment of $\mathrm{Ho}(M)$ over $\mathrm{Ho}(V)$ satisfies these axioms.

Has anyone done this? Is it useful for anything?

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I haven't seen it, and I'd say it could be useful, although I don't have any concrete application in mind. – Fernando Muro Jan 22 '14 at 12:15

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