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 seems true in nice situations; e.g. if $(\mathbf{C}, \mathbf{W})$ has a calculus of fractions, or if it has a model structure (or more generally a 3-arrow calculus), but I'm curious about the general situation, and have had no success working out a proof or a counterexample.

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

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

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; e.g. if $(\mathbf{C}, \mathbf{W})$ has a calculus of fractions, or if it has a model structure (or more generally a 3-arrow calculus), but I'm curious about the general situation, and have had no success working out a proof or a counterexample.

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

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 seems true in nice situations; e.g. if $(\mathbf{C}, \mathbf{W})$ has a calculus of fractions, or if it has a model structure (or more generally a 3-arrow calculus), but I'm curious about the general situation, and have had no success working out a proof or a counterexample.

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