# Bousfield localization before and after taking homotopy

The ncatlab says:

Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left Bousfield localization of $C$ is the left Bousfield localization of $\mathrm{Ho}(C)$.

I wholeheartedly agree. Does anyone know of results in this direction?

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## 1 Answer

Let $\tilde C$ be the left Bousfield localization of $C$. As categories $C=\tilde C$, but $\tilde C$ has more weak equivalences. In particular, the identity functor induces a functor $\varphi\colon \operatorname{Ho}(C)\rightarrow \operatorname{Ho}(\tilde C)$. Let $\mathcal L=\ker \varphi$, i.e. $\mathcal L\subset \operatorname{Ho}(C)$ is the full subcategory spanned by the objects which become trivial in $\operatorname{Ho}(\tilde C)$. If $\tilde C$ is stable and $\varphi$ preserves homotopy colimits (and these are the suitable conditions, I think) then $\varphi$ is an exact functor between triangulated categories, so $\mathcal L$ is a thick subcategory of $\operatorname{Ho}(C)$ and $\varphi$ induces a functor from the Verdier quotient $\bar\varphi\colon\operatorname{Ho}(C)/\mathcal L\rightarrow \operatorname{Ho}(\tilde C)$. Let us show that $\bar\varphi$ is an equivalence of categories. It is enough to prove that the canonical composition of 'projection' functors $\tilde C=C\rightarrow\operatorname{Ho}(C)\rightarrow \operatorname{Ho}(C)/\mathcal L$ satisfies the universal property of $\tilde C\rightarrow \operatorname{Ho}(\tilde C)$. Suppose $\psi\colon C\rightarrow D$ is a functor which sends all weak equivalences in $\tilde C$ to isomorphisms. Since weak equivalences in $C$ are weak equivalences in $\tilde C$, $\psi$ factors through $C\rightarrow \operatorname{Ho}(C)$ in an essentially unique way. Let $\bar\psi\colon \operatorname{Ho}(C)\rightarrow D$ be the factorizaton. Recall that $\operatorname{Ho}(C)/\mathcal L$ is the localization of $\operatorname{Ho}(C)$ inverting those maps whose mapping cone is in $\mathcal L$. Any such map is represented by a zig-zag of weak equivalences in $\tilde C$. Since $\psi$ sends weak equivalences in $\tilde C$ to isomorphisms then $\bar\psi$ factors through $\operatorname{Ho}(C)\rightarrow \operatorname{Ho}(C)/\mathcal L$ in an essentially unique way. In particular $\psi$ factors through the composite $\tilde C\rightarrow \operatorname{Ho}(C)/\mathcal L$. The essential uniqueness of this factorization follows from the aforementioned essential uniqueness of the two intermediate steps. I hope this argument is correct and convincing, and that I haven't missed any necessary hypothesis.

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Thanks a lot! I should have tried more seriously to see on my own if one can just prove this in a straight-forward manner. I guess the nlab entry made me think this was non-obvious and unknown. Is it okay if I put a link to this answer in the nlab entry? –  Rasmus Bentmann Jul 12 '13 at 15:08
You're welcome! You can link to this answer, or expand it conveniently. I think it's a little bit sketchy, isn't it? –  Fernando Muro Jul 12 '13 at 15:20
I will put a link for now. In the future I will hopefully replace it with a more detailed version of your answer. –  Rasmus Bentmann Jul 12 '13 at 15:24
I am a bit puzzled by the condition that $\varphi$ preserve homotopy colimits because it seems to be a condition on the output data $\mathrm{Ho}(C)\to\mathrm{Ho}(\tilde C)$ and not (explicitly) on the input data $(C,\tilde C)$ (modulo the fact that there are no functorial homotopy (co)fibers in triangulated categories). Also, would it not be enough to have that $\varphi$ preserves finite coproducts and homotopy (co)fibers to get additivity and exactness, respectively? –  Rasmus Bentmann Aug 10 '13 at 8:55
Preserving finite homotopy coproducts and homotopy cofibers is the same as preserving finite homotopy colimits, which is enough. As you said, this condition is really a condition on the model categories, or if you wish on the induced derivators, but not on the homotopy categories. I abused terminology. –  Fernando Muro Aug 10 '13 at 12:36