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Rings and Modules in Kan Spectra

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    R. Chen & I. Kriz

  2. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    A. Pultr

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  1. R. Chen
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  2. I. Kriz
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Correspondence to I. Kriz.

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Communicated by Maria Manuel Clementino.

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Chen acknowledges support of NSF RTG Grant 1045119. Kriz acknowledges the support of a Simons Collaboration Grant. Pultr extends his thanks to KAM, Faculty of Mathematics and Physics of the Charles University in Prague.

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Chen, R., Kriz, I. & Pultr, A. Rings and Modules in Kan Spectra. Appl Categor Struct 31, 18 (2023). https://doi.org/10.1007/s10485-023-09719-y

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