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Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}\to X_n), (q_{n+1}: Y_{n+1}\to Y_n), (f_n: X_n\to Y_n), n=0, 1,\ldots$ with $q_{n+1}f_{n+1}=f_np_{n+1}$, so we get a ladder of commutative squares. If each $p_n$ is a fibration of fibrants in $M$, each $q_n$ is a fibration of fibrants in $L_CM$, and each $f_n$ is a weak equivalence in $L_CM$, can we conclude that the limit map $\lim f_n$ is also a weak equivalence in $L_CM$?

For the notion of left Bousfield localization, see Hirschhorn, Model categories and their localizations, chapter 3, 4. See Proposition 15.10.12 in that book for a similar result, my question is by weakening the assumption as well as the conclusion. You may add suitable and reasonable conditions—like simplicial, properness, cofibrantly generated, etc.—if needed.

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No. For a counterexample to your claim, consider the model category M of simplicial presheaves on a small site S equipped with the projective model structure. Its fibrant objects are presheaves of Kan complexes. If C is the set of Čech covers of S, then L_C(M) is the local projective model structure on simplicial presheaves. Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent. A weak equivalence from a fibrant object in M to a fibrant object in L_C(M) is a homotopy sheafification map. Furthermore, the limit of p and q is a homotopy limit in M, so lim f_n is a weak equivalence if and only if the homotopy sheafification functor preserves homotopy limits of towers. This is false for arbitrary sites.

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    $\begingroup$ What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $\lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def? $\endgroup$ – Lao-tzu Dec 26 '18 at 11:38
  • $\begingroup$ The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $\infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations. $\endgroup$ – Harry Gindi Dec 26 '18 at 12:32
  • $\begingroup$ @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $\infty$-categories (when taking the $\infty$-category associated to the given model category) or there is some relation you can explain? $\endgroup$ – Lao-tzu Dec 26 '18 at 13:11
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    $\begingroup$ @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram. $\endgroup$ – Harry Gindi Dec 26 '18 at 15:31
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    $\begingroup$ @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion. $\endgroup$ – Dmitri Pavlov Dec 26 '18 at 15:38
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In the language of $\infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $\infty$-categories.

Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).

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    $\begingroup$ As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.) $\endgroup$ – Harry Gindi Dec 26 '18 at 5:35

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