2022/06/02 by Paul Lessard, Lessard, Paul · 1 voice
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #math.AT #math.CT
paper · pdf · doi:10.48550/arxiv.2206.00849
openalex publication_date 2022/06/02 · arxiv published 2022/06/02 · arxiv updated 2022/06/02 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
This paper is the first in a series of two papers, Z-Categories I and Z-Categories II, which develop the notion of Z-category, the natural bi-infinite analog to strict ω-categories, and show that the (∞,1)-category of spectra relates to the (∞,1)-category of homotopy coherent Z-categories as the pointed groupoids. In this work we provide a 2-categorical treatment of the combinatorial spectra of \citeKan and argue that this description is a simplicial avatar of the abiding notion of homotopy coherent 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 Z-categories as \triangle is to 1-categories or Θn is to n-categories. In an appendix we provide a generalization of the spectrification functors of 20th century stable homotopy theory in the language of category-weighted limits.