Kan's combinatorial spectra and their sheaves revisited

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.