Model structure for fiberwise Bousfield localization

Question

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.

Answer 1

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.