Let $S$ be a small family of arrows in a locally presentable category $\mathcal{K}$.

It is known that the category $\mathcal{K}[S^{-1}]$ is reflective in $\mathcal{K}$ and correspond to the solution of the orthogonality problem associated to $S$.

Can I infer that a map is an iso in $\mathcal{K}[S^{-1}]$ if and only if it comes from a map in $S$?

This question is strongly related to this other one. In that case, the localization might not be reflective, and this difference might be quite relevant in the answer.


No, you cannot infer this. The class of morphisms inverted by any functor must satisfy the two-for-three, even the two-for-six, property. But $S$ is completely arbitrary. Consider, for a dramatic example, $\mathrm{Set}[(\emptyset \to \{*\})^{-1}]$, where inverting a single arrow is equivalent to inverting all arrows. Furthermore, there are many more conditions than just two-for-$n$ in this situation. Indeed, the class of morphisms inverted by any left adjoint is closed under colimits in the arrow category.


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