Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) \times \operatorname{Ho} (\cal D)$, and that this is true for any (finite I assume) number of model categories.


Is this true in general? Let $\{ \mathcal C\_i \}\_{i \in I}$ be a family of categories, let $W\_i \subseteq \operatorname{Mor} \cal C\_i$ be families of morphisms and let $\mathcal C = \prod\_{i \in I} \cal C\_i$. Are the localisations $$\mathcal C\left[ \prod\_{i \in I} W\_i ^{-1}\right] \cong \prod\_{i \in I} \mathcal C\_i\left[W\_i^{-1}\right]?$$


I can find a functor from the left to the right in general.


Is it at least true for infinite families of model categories?



Is the localisation of a product of categories the product of the localisation?
It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of <http://arxiv.org/abs/math/0610009v4>.


It is also true for finite products of arbitrary relative categories as discussed in this answser: [Localizing an arbitrary additive category](https://mathoverflow.net/questions/44047/localizing-an-arbitrary-additive-category/44155#44155). (For more details on this and related arguments see <http://nforum.mathforge.org/discussion/4769/connected-components-preserve-finite-products/>.)


**Addendum:** This doesn't hold for infinite products of arbitrary relative categories. Consider a sequence of relative categories $\mathcal{C}\_0, \mathcal{C}\_1, \mathcal{C}\_2, \ldots$ such that $\mathcal{C}\_i$ is saturated and has objects $X\_i$ and $Y\_i$ that are weakly equivalent, but the shortest zig-zag of weak equivalences witnessing this is at least $i$ arrows long. Then $(X\_0, X\_1, \ldots)$ and $(Y\_0, Y\_1, \ldots)$ are isomorphic as objects of $\prod\_i \mathrm{Ho}(\mathcal{C}\_i)$ but not as objects of $\mathrm{Ho}(\prod\_i \mathcal{C}\_i)$ since this would imply that the length of the shortest zig-zag connecting $X\_i$ to $Y\_i$ is bounded.


