# Model structure for fiberwise Bousfield localization

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.

• Probably an easy question: should we ever expect to have a model structure on path connected things? It would seem to me like path objects in any interesting structure will generally not be connected. – Connor Malin Nov 11 '19 at 15:38
• We don't need this condition, now edited. – Manuel Rivera Nov 11 '19 at 15:45
• This isn't quite what you asked for, but: given a fixed base space $B$, one has a model structure on the category of spaces over $B$. Call a map $E \to E'$ over $B$ a fiberwise $R$-equivalence if the induced map on homotopy pullbacks $F \to F'$ as an $R$-homology equivalence. Then, relative to $B$, you can ask for a fiberwise model structure; essentially by definition it's the left Bousfield localization with respect to fiberwise $R$-equivalences. In particular, one checks that the fibrant objects are fibrations $E \to B$ with $R$-local fibers. – Tyler Lawson Nov 11 '19 at 15:54
• There are various base-change functors between these model categories as $B$ varies; this is less convenient in some sense, but it also frees you from being dependent on path-connectivity or from only looking at the fundamental group of $X$. E.g.: if $\pi_1(X)$ has a perfect normal subgroup $P$, then Quillen's plus-construction is a fiberwise $\Bbb Z$-localization of $X \to K(\pi_1(X)/P,1)$. – Tyler Lawson Nov 11 '19 at 15:57
• hi @TylerLawson, yes I agree, but this is different. Its a bit confusing because this is what Hirschhorn calls fiberwise localization in his book I think, while I meant something different. – Manuel Rivera Nov 11 '19 at 15:58

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.

• Thanks, this makes sense. Sorry, I was sloppy with the connected components in my original question. The argument seems to go through for simplicial sets with one vertex as well. Anyway, I thought this model structure was studied and used in some of Bousfield Kan papers. For example, it leads to the non-simply connected version of rational homotopy theory studied here: ams.org/journals/tran/2000-352-04/S0002-9947-99-02463-0/… – Manuel Rivera Nov 11 '19 at 21:54