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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?

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Continued at where I also gave a complete answer for the case that $I$ is finite. – Martin Brandenburg Aug 15 '13 at 14:51
up vote 12 down vote accepted

It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of

It is also true for finite products of arbitrary relative categories as discussed in this answser: Localizing an arbitrary additive category. (For more details on this and related arguments see

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.

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