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

TL;DR

This paper explores the relationships between cores and localizations of $( abla, abla)$-categories as the dimension tends to infinity, revealing their structural connections and intermediate localizations.

Contribution

It compares core and localization functors for $( abla,d)$-categories as $d o abla$, showing the localization-limit is a reflective localization of the core-limit and analyzing intermediate notions of invertibility.

Findings

Localization-limit is a reflective localization of the core-limit

Intermediate localizations arise from notions of invertibility at infinity

Comparison enhances understanding of $( abla, abla)$-category structures

Abstract

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.