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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?

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  • $\begingroup$ 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 $S_L$? $\endgroup$
    – Valery Isaev
    Commented Jan 16, 2020 at 5:20
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    $\begingroup$ Is this the same notion of fiberwise localization that appears in part I of Hirschhorn's book on model categories? $\endgroup$
    – Mike Shulman
    Commented Jan 16, 2020 at 18:14
  • $\begingroup$ @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. $\endgroup$
    – Tim Campion
    Commented Jan 16, 2020 at 22:08
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    $\begingroup$ $E \to \bar E \to B$ 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. $\endgroup$
    – Valery Isaev
    Commented Jan 17, 2020 at 4:42
  • $\begingroup$ I know that Vandembroucq proved a uniqueness theorem for fiberwise localizations of unpointed functors in MR1923222 $\endgroup$
    – Jeff Strom
    Commented Jan 21, 2020 at 15:20

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