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

Viktoriya Ozornova, Martina Rovelli, Tashi Walde

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.

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

Abstract

We consider -categories in the limit via the core or localization functors that forget or invert higher non-invertible arrows, respectively. We compare the two resulting -categories of -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 such as the one defined by coinduction.
Paper Structure (22 sections, 59 theorems, 78 equations)

This paper contains 22 sections, 59 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. What is an $(\infty,\infty)$-category?
  3. $(\infty,\infty)$-categories via cores or localizations
  4. External localizations via internal invertibility
  5. Acknowledgements
  6. Preliminaries
  7. Animae and categories
  8. Enriched categories
  9. $d$-categories
  10. Localizations and cores
  11. Surjectivity
  12. Surjectivity and localization
  13. Biinductive systems
  14. ... with a bias
  15. Left and right $\infty$-categories
  16. ...and 7 more sections

Key Result

Theorem 2.7

Theorems & Definitions (182)

  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: GH, Definition 4.3.1
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8
  • ...and 172 more