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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 26th 2014

    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?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeAug 26th 2014

    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.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 26th 2014

    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).

    • CommentRowNumber4.
    • CommentAuthorZhen Lin
    • CommentTimeAug 26th 2014

    Here’s a fairly straightforward special case. Let 1\mathcal{M}_1 and 2\mathcal{M}_2 be model categories where all objects in 1\mathcal{M}_1 are cofibrant and suppose we have a left Quillen functor F: 1 2F : \mathcal{M}_1 \to \mathcal{M}_2. Then for any sets S 1mor 1S_1 \subseteq \operatorname{mor} \mathcal{M}_1 and S 2mor 2S_2 \subseteq \operatorname{mor} \mathcal{M}_2, if FF sends morphisms in S 1S_1 to S 2S_2, then FF is also a left Quillen functor L S 1 1L S 2 2L_{S_1} \mathcal{M}_1 \to L_{S_2} \mathcal{M}_2. Indeed, if G: 2 1G : \mathcal{M}_2 \to \mathcal{M}_1 is the right adjoint, then GG sends (fibrant) S 2S_2-local objects in 2\mathcal{M}_2 to (fibrant) S 1S_1-local objects in 1\mathcal{M}_1, so FF must send S 1S_1-local equivalences in 1\mathcal{M}_1 to S 2S_2-local equivalences in 2\mathcal{M}_2.

    In particular, if 1\mathcal{M}_1 and 2\mathcal{M}_2 have the same underlying category, F=idF = id, S 1=S 2S_1 = S_2, and all objects are cofibrant, then this says S 1S_1-equivalences are S 2S_2-equivalences.