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

TL;DR

This paper introduces a new $ ext{(} ext{infinity,} ext{Z} ext{)}$-category framework, connecting categorical spectra with weak $ ext{Z}$-categories, and clarifies their stable cell structures.

Contribution

It provides an $ ext{(} ext{infinity,} ext{Z} ext{)}$-categorical definition of weak $ ext{Z}$-categories and proves their equivalence to categorical spectra, extending the understanding of spectra in higher category theory.

Findings

Categorical spectra are equivalent to pointed $( ext{infinity,} ext{Z})$-categories.

Stable cells of categorical spectra match natural cells of $( ext{infinity,} ext{Z})$-categories.

Recovers Lessard's description of spectra as pointed weak $ ext{Z}$-groupoids.

Abstract

Lessard's $\mathbb{Z}$-categories are an analogue of $\omega$-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 $(\infty,\omega)$-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.