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.



Limit of weak equivalences in a Bousfield localization
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.


