Rings and Modules in Kan Spectra

The purpose of this paper is to set up derived categories of sheaves of \(E_\infty \)-rings and modules over non-derived sites, in particular over topological spaces. This theory opens up certain new capabilities in spectral algebra. For example, as outlined in the last section of the present paper, using these concepts, one can conjecture a spectral algebra-based generalization of the geometric Langlands program to manifolds of dimension \(>2\). As explained in a previous paper (Chen et al. in Theory Appl Categ 32:1363-1396, 2017) the only theory of sheaves of spectra on non-derived sites known to date which has well-behave pushforwards is based on Kan spectra, which, however, are reputed not to possess a smash product rigid enough for discussing \(E_\infty \)-objects. The bulk of this paper is devoted to remedying this situation, i.e. defining a more rigid smash product of Kan spectra, and using it to construct the desired derived categories.