I think the following should be in the literature but couldn't find it. 


Recall that around the 1970's Bousfield described the $R$-localization $EX$ of any space $X$, for $R$ a fixed ring. The construction comes with a natural map $X \to EX$, which induces an isomorphism $H\_\*(X;R) \cong H\_\*(EX;R)$ and satisfies a universal property.


Then he constructed a model category structure on the category of simplicial sets for which weak equivalences are maps that induce isomorphisms on $H\_\*( \text{ }; R)$ and the cofibrations are injections. 


Later on, Bousfield and Kan, as well as E. Dror Farjoun, described a *fiberwise localization* which, given any pointed space $(X,b)$, may be applied to the fibration $\tilde{X} \to X \to B\pi\_1(X,b)$, where $\tilde{X}$ is the universal cover of $X$, to (functorially) produce a new fibration $E\tilde{X} \to E'X \to B\pi\_1(X,b)$ for which the fibers are the $R$-localization of $\tilde{X}$.


Question: Is there a model category structure on pointed spaces with the same cofibrations as above and whose weak equivalences are maps $X \to Y$ for which the induced map $E'X \to E'Y$ is a weak homotopy equivalence of fibrations?


I'd appreciate if someone could point out a reference. I couldn't find it explicitly in the work of Bousfield, Kan, or Dror-Farjoun, but it should be in the literature. It should also hold in more general contexts. 



Model structure for fiberwise Bousfield localization
Here are two variants on this.


Strictly, there is no such model structure. If $X$ is any space, then the map $X\_+ \to CX\_+$ is an acyclic cofibration under the definitions given, where $CX$ is the cone on $X$: the definition of equivalence does not see components away from the basepoint. Taking the pushout of the diagram $X \leftarrow X\_+ \to CX\_+$, the map $X \to CX$ would then have to be an acyclic cofibration. (Perhaps a version with reduced simplicial sets, as Jeff Strom suggests, would get around this issue).


However, a basepoint-free way to proceed might be to say that a map $f: X \to Y$ of simplicial sets is a weak equivalence if:


* it is an isomorphism on $\pi\_0$;
* for any basepoint $b \in X$, the map $\pi\_1(X,b) \to \pi\_1(Y,f(b))$ is an isomorphism; and
* for any basepoint $b \in X$, the induced map $\tilde X \to \tilde Y$ of universal covers at that basepoint is an $R$-homology isomorphism.


This class of $W$ weak equivalences satisfies the following properties:


* it satisfies the 2-out-of-3 axiom;
* it is implied by being a weak equivalence of spaces, so acyclic fibrations are in $W$;
* cofibrations in $W$ are closed under pushout and transfinite composition; and
* it satisfies the hypotheses of the Bousfield-Smith cardinality argument. The acyclic cofibrations can be generated by (a) ordinary acyclic cofibrations of simplicial sets, and (b) cofibrations $A \to B$ between simply-connected spaces which are $R$-homology isomorphisms. Both classes are generated by a set; the former by the standard generating cofibrations, and Bousfield's argument shows that the latter is generated by a set as well.


Smith's theorem then says that there is a model structure with these cofibrations and weak equivalences; it's a left Bousfield localization of the standard model structure on simplicial sets. The effect of fibrant replacement is to $R$-complete the universal cover at each path component.


Unfortunately I do not know of a reference for you; my apologies.


