Categorical spectra as pointed $(\infty ,\mathbb{Z})$-categories

Abstract

Lessard's $\mathbb{Z}$-categories are an analogue of $\ddot{I}‰$-categories possessing cells in all positive and negative dimensions. Categorical spectra, developed by Stefanich, are an analogue of spectra obtained by replacing the suspension of pointed $\infty$-groupoids by that of pointed $(\infty ,\ddot{I}‰)$-categories. We give an $\infty$-categorical definition of weak $\mathbb{Z}$-categories (alias $(\infty ,\mathbb{Z})$-categories), and show categorical spectra to be equivalent to pointed $(\infty ,\mathbb{Z})$-categories. In particular, we show that the stable cells of categorical spectra coincide with the natural cells of $(\infty ,\mathbb{Z})$-categories, and recover Lessard's description of spectra as pointed weak $\mathbb{Z}$-groupoids.

16 pages. Comments are welcome! V2: Added section (now) 3.2 on monoidal structures