# Does Farjoun's “fiberwise localization” have a universal property?

Let $$\mathcal S_L$$ be any accessible reflective subcategory of the $$\infty$$-category of spaces. In his book, Farjoun discusses "fiberwise $$L$$-localization" of a map of spaces $$E \to B$$, i.e. a factorization $$E \to \bar E \to B$$ such that the fiber of $$\bar E \to B$$ is in $$\mathcal S_L$$ and $$E \to \bar E$$ is $$\mathcal S_L$$-local. Chataur and Scherer show (Thm 4.3) that a fiberwise localization always exists.

Farjoun claims (paragraph 1.F.5) that any fiberwise localization enjoys a universal property, but he's not quite precise on what that means -- he says (with slight paraphrase) "the map $$E \to \bar E$$ as a map over $$B$$ is universal among all maps of $$E$$ to spaces over $$B$$ that have homotopy fiber in $$\mathcal S_L$$".

I'm not quite clear on what "universal" means here, but surely it should at least imply that $$\bar E$$ is unique up to equivalence over $$B$$. But I'm having trouble seeing that. So I ask,

Question: Let $$\mathcal S_L$$ be any accessible reflective subcategory of the $$\infty$$-category of spaces. Let $$E \to B$$ be a map and let $$\bar E \to B$$, $$\bar E' \to B$$ be two fiberwise $$L$$-localizations of $$E \to B$$. Then are $$\bar E \to B$$, $$\bar E' \to B$$ equivalent over $$B$$ (and under $$E$$)? More strongly, is the space of fiberwise localizations of $$E \to B$$ contractible?

• Isn't $E \to \bar E \to B$ just the initial object in the $\infty$-category of factorizations of $E \to B$ in which the second map is in $S_L$? – Valery Isaev Jan 16 at 5:20
• Is this the same notion of fiberwise localization that appears in part I of Hirschhorn's book on model categories? – Mike Shulman Jan 16 at 18:14
• @ValeryIsaev That's what I'd like to be true, but I'm not sure that actually follows from the definition -- i.e. from the condition that the fiber of $\bar E \to B$ be in $\mathcal S_L$ and $E \to \bar E$ be $\mathcal S_L$-local. – Tim Campion Jan 16 at 22:08
• $E \to \bar E \to B$ has the universal property with respect to factorizations satisfying stronger property that the second map is $S_L$-local in the category over $B$ (where I assume that $S_L$ is define as localization with respect to a set of maps). It is not enough to require that its fiber is $S_L$-local. For example, the fiber of the Hopf fibration $f : S^3 \to S^2$ is $f$-local, but the actual $f$-localization of $f$ over $S^2$ is the identity map. – Valery Isaev Jan 17 at 4:42
• I know that Vandembroucq proved a uniqueness theorem for fiberwise localizations of unpointed functors in MR1923222 – Jeff Strom Jan 21 at 15:20