Kan's combinatorial spectra and their sheaves revisited

TL;DR

This paper establishes a model structure on Kan's combinatorial spectra and their sheaves, using geometric homotopy concepts, as a foundational step for geometric analysis of sheaves of spectra.

Contribution

It introduces a right Cartan-Eilenberg structure on combinatorial spectra and sheaves, emphasizing geometric homotopy equivalences, advancing the study of sheaves of spectra from a geometric perspective.

Findings

Defined a model structure on combinatorial spectra

Established a model structure on sheaves of spectra

Used geometric homotopy as the core equivalence concept

Abstract

We define a right Cartan-Eilenberg structure on the category of Kan's combinatorial spectra, and the category of sheaves of such spectra, assuming some conditions. In both structures, we use the geometric concept of homotopy equivalence as the strong equivalence. In the case of sheaves, we use local equivalence as the weak equivalence. This paper is the first step in a larger-scale program of investigating sheaves of spectra from a geometric viewpoint.