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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 26th 2014
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   Author: Dmitri Pavlov
   Format: MarkdownItexConsider a left proper combinatorial model category C and a set of morphisms S in C. Weak equivalences in the left Bousfield localization of C with respect to S are precisely those morphisms that are inverted by the Dwyer-Kan localization of C with respect to the union of S and weak equivalences of C. Is there a written reference for this fact?

   Consider a left proper combinatorial model category C and a set of morphisms S in C.

   Weak equivalences in the left Bousfield localization of C with respect to S are precisely those morphisms that are inverted by the Dwyer-Kan localization of C with respect to the union of S and weak equivalences of C.

   Is there a written reference for this fact?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeAug 26th 2014
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   Author: Zhen Lin
   Format: MarkdownItexIs it even true? It's very close to claiming this: every localisation of a locally presentable category with respect to a small set of morphisms is a reflective localisation. But _that_ is not true.

   Is it even true? It’s very close to claiming this: every localisation of a locally presentable category with respect to a small set of morphisms is a reflective localisation. But that is not true.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 26th 2014
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   Author: Dmitri Pavlov
   Format: MarkdownItexI guess I really meant accessible / reflective localizations. Is there such a thing as an accessible or reflective Dwyer-Kan localization? The reason I'm interested in such a statement is a monotonicity property for left Bousfield localizations: if W₁ ⊂ W₂ are two model structures on the same category and S is a set of morphisms, then weak equivalences of L_S(W₁) are contained in weak equivalences of L_S(W₂).

   I guess I really meant accessible / reflective localizations. Is there such a thing as an accessible or reflective Dwyer-Kan localization?

   The reason I’m interested in such a statement is a monotonicity property for left Bousfield localizations: if W₁ ⊂ W₂ are two model structures on the same category and S is a set of morphisms, then weak equivalences of L_S(W₁) are contained in weak equivalences of L_S(W₂).

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeAug 26th 2014
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   Author: Zhen Lin
   Format: MarkdownItexHere's a fairly straightforward special case. Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be model categories where all objects in $\mathcal{M}_1$ are cofibrant and suppose we have a left Quillen functor $F : \mathcal{M}_1 \to \mathcal{M}_2$. Then for any sets $S_1 \subseteq \operatorname{mor} \mathcal{M}_1$ and $S_2 \subseteq \operatorname{mor} \mathcal{M}_2$, if $F$ sends morphisms in $S_1$ to $S_2$, then $F$ is also a left Quillen functor $L_{S_1} \mathcal{M}_1 \to L_{S_2} \mathcal{M}_2$. Indeed, if $G : \mathcal{M}_2 \to \mathcal{M}_1$ is the right adjoint, then $G$ sends (fibrant) $S_2$-local objects in $\mathcal{M}_2$ to (fibrant) $S_1$-local objects in $\mathcal{M}_1$, so $F$ must send $S_1$-local equivalences in $\mathcal{M}_1$ to $S_2$-local equivalences in $\mathcal{M}_2$. In particular, if $\mathcal{M}_1$ and $\mathcal{M}_2$ have the same underlying category, $F = id$, $S_1 = S_2$, and all objects are cofibrant, then this says $S_1$-equivalences are $S_2$-equivalences.

   Here’s a fairly straightforward special case. Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be model categories where all objects in $\mathcal{M}_1$ are cofibrant and suppose we have a left Quillen functor $F : \mathcal{M}_1 \to \mathcal{M}_2$. Then for any sets $S_1 \subseteq \operatorname{mor} \mathcal{M}_1$ and $S_2 \subseteq \operatorname{mor} \mathcal{M}_2$, if $F$ sends morphisms in $S_1$ to $S_2$, then $F$ is also a left Quillen functor $L_{S_1} \mathcal{M}_1 \to L_{S_2} \mathcal{M}_2$. Indeed, if $G : \mathcal{M}_2 \to \mathcal{M}_1$ is the right adjoint, then $G$ sends (fibrant) $S_2$-local objects in $\mathcal{M}_2$ to (fibrant) $S_1$-local objects in $\mathcal{M}_1$, so $F$ must send $S_1$-local equivalences in $\mathcal{M}_1$ to $S_2$-local equivalences in $\mathcal{M}_2$.

   In particular, if $\mathcal{M}_1$ and $\mathcal{M}_2$ have the same underlying category, $F = id$, $S_1 = S_2$, and all objects are cofibrant, then this says $S_1$-equivalences are $S_2$-equivalences.