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Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial localization $L(C^D,W^D)$. Or I can first simplicially localize to get $L(C,W)$ and then form the $(\infty,1)$-functor category $L(C,W)^D$.

If $(C,W)$ is a nice model category, so that $(C^D,W^D)$ has a model structure presenting $L(C,W)^D$, then we have $L(C^D,W^D) \simeq L(C,W)^D$. Karol gave an easy counterexample of $C,W,D$ (without a model structure) for which it fails, depending on the fact that in general, morphisms in $L(C,W)$ may require zigzags of arbitrarily long length in $C$. But what if we assume some structure on $C$ less than a model category, but sufficient to ensure that zigzags of bounded length suffice?

For concreteness, how about if we assume a 3-arrow calculus as defined in the DHKS book Homotopy limit functors on model categories and homotopical categories? Recall that this means we have subcategories $U,V$ of $C$ satisfying the "functorial left (right) Ore condition" respectively and such that every map in $W$ factors functorially as one in $U$ followed by one in $V$. This ensures that every morphism in $L(C,W)$ can be represented by a 3-term zigzag, like in a model category.

I'd also be interested in answers to similar questions, such as what if $W$ admits a left or right calculus of fractions.

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  • $\begingroup$ If the answer is yes, it would appear to say that homotopy-coherent diagrams can be rectified to strict diagrams. (I previously asked a question along those lines.) In particular, if this is not possible for the given $(C, W)$ and $D$, then one cannot have $L (C^D, W^D) \simeq L(C, W)^D$. $\endgroup$
    – Zhen Lin
    Jul 8, 2014 at 0:25
  • $\begingroup$ Clark Barwick would seem to be the person to ask about this: the techniques of <arxiv.org/abs/1011.1691> and it's sequels ought to be useful here. $\endgroup$ Jul 28, 2014 at 22:00
  • $\begingroup$ @ClarkBarwick, any thoughts? $\endgroup$ Jul 29, 2014 at 18:17
  • $\begingroup$ What is a “nice” model category in the original post and how does one prove the statement about localization for the case of model categories? $\endgroup$ Oct 2, 2014 at 11:42

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