Limit of weak equivalences in a Bousfield localization 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.
 A: 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).
A: 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.
