$\mathbf{Z}$-Categories I

TL;DR

This paper introduces the concept of $ extbf{Z}$-categories as a bi-infinite analog of strict $oldsymbol{ extomega}$-categories, connecting spectra and homotopy coherent $ extbf{Z}$-categories through a 2-categorical framework.

Contribution

It develops the theory of $ extbf{Z}$-categories, providing a 2-categorical perspective and a cellular structure analogous to classical category theory, and generalizes spectrification functors.

Findings

Establishes a 2-categorical treatment of combinatorial spectra.

Defines a cellular category for $ extbf{Z}$-categories analogous to $ riangle$ and $ heta_n$.

Generalizes spectrification functors using category-weighted limits.

Abstract

This paper is the first in a series of two papers, $\mathbf{Z}$-Categories I and $\mathbf{Z}$-Categories II, which develop the notion of $\mathbf{Z}$-category, the natural bi-infinite analog to strict $\omega$-categories, and show that the $\left(\infty,1\right)$-category of spectra relates to the $\left(\infty,1\right)$-category of homotopy coherent $\mathbf{Z}$-categories as the pointed groupoids. In this work we provide a $2$-categorical treatment of the combinatorial spectra of \cite{Kan} and argue that this description is a simplicial avatar of the abiding notion of homotopy coherent $\mathbf{Z}$-category. We then develop the theory of limits in the $2$-category of categories with arities of Berger, Mellies, and Weber to provide a cellular category which is to $\mathbf{Z}$-categories as $\triangle$ is to $1$-categories or $\Theta_{n}$ is to $n$-categories. In an appendix we provide a generalization of the spectrification functors of 20$^{\mathrm{th}}$ century stable homotopy theory in the language of category-weighted limits.