Cores and localizations of $(\infty,\infty)$-categories

We consider $(\infty,d)$-categories in the limit $d\to \infty$ via the core or localization functors that forget or invert higher non-invertible arrows, respectively. We compare the two resulting $(\infty,1)$-categories of $(\infty,\infty)$-categories and exhibit the localization-limit as a reflective localization of the core-limit. On the side, we study intermediate localizations that arise from notions of invertibility that only emerge at $d=\infty$ such as the one defined by coinduction.