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Let $(\mathbf{C}, \mathbf{W})$ be a relative category where $\mathbf{W}$ is saturated; i.e. it is equal to the subcategory of arrows inverted by $\gamma : \mathbf{C} \to \mathbf{C}[\mathbf{W}^{-1}]$.

The core of a category is the subcategory of all invertible arrows. $\gamma$ induces a functor $\mathbf{W}[\mathbf{W}^{-1}] \to \mathrm{Core}(\mathbf{C}[\mathbf{W}^{-1}])$.

Question 1: Is $\mathbf{W}[\mathbf{W}^{-1}] \to \mathrm{Core}(\mathbf{C}[\mathbf{W}^{-1}])$ an equivalence of groupoids?

This is true in nice situations, such as if $(\mathbf{C}, \mathbf{W})$ can be further equipped with a model structure. However, I'm curious about the general case, and have had no success working out a proof or a counterexample.

The saturation condition cannot be omitted; it's not hard to construct counterexamples if $\mathbf{W}$ isn't required to contain all arrows that are mapped to isomorphisms by $\gamma$.

I am interested in the answer to the same question in the more general contexts:

Question 2: The same as question 1, but the localizations are in the sense of $\infty$-categories

Question 3: The same as question 2, but $\mathbf{C}$ is a general $\infty$-category

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    $\begingroup$ Cool name, brother. $\endgroup$
    – Squid with Black Bean Sauce
    Commented Dec 2, 2019 at 20:53
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    $\begingroup$ @DenisNardin Post that as an answer? $\endgroup$
    – Mike Shulman
    Commented Dec 3, 2019 at 4:10
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    $\begingroup$ Sorry, I had missed the saturation hypothesis on $W$, ignore what I wrote. $\endgroup$
    – Denis Nardin
    Commented Dec 3, 2019 at 9:20
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    $\begingroup$ @EdoardoLanari: The isos in $\mathbf{C}$ are mapped to isos in $\mathbf{C}[W^{-1}]$, and thus by the saturation condition, they are already isos in $\mathbf{W}$. $\endgroup$
    – CoreLocalization
    Commented Dec 3, 2019 at 15:38
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    $\begingroup$ It is true for $\infty$-categories with fibrations, see Corollary 7.6.9 of Cisinski's book. $\endgroup$
    – Andrea Gagna
    Commented Dec 3, 2019 at 17:02

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