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Do kernels of heart-exact localizations characterize c-categories?

Determine whether every c-category can be realized as the kernel of a heart-exact localization from a bounded weighted category to a finite-cohomological-dimension (fcd) heart category. Concretely, decide if “being the kernel of a heart-exact localization from a bounded weighted category to an fcd heart category” characterizes c-categories; the authors suspect this is false.

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Background

The paper shows c-categories admit functorial resolutions by bounded weighted categories and discusses kernels of localizations in constructing such resolutions.

While many c-categories arise as such kernels, the authors are unsure if this provides an exact characterization of all c-categories.

References

We do not know (and suspect it is false) that being the kernel of a heart exact localization from a bounded weighted category to an fcd heart category characterizes $c$-categories. See \Cref{question:kerlocccat} and \Cref{question:width0}, and see \Cref{thm:weakimpliesstrong} for a characterization of such kernels.

— $c$-structures and trace methods beyond connective rings (2509.14774 - Levy et al., 18 Sep 2025) in Remark “characterization,” Section 3 (Resolving c-categories)