{"title":"Rings and Modules in Kan Spectra","authors":"R. Chen, I. Kriz, A. Pultr","doi":"10.1007/s10485-023-09719-y","DOIUrl":null,"url":null,"abstract":"<div><p>The purpose of this paper is to set up derived categories of sheaves of <span>\\(E_\\infty \\)</span>-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 <span>\\(>2\\)</span>. 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 <span>\\(E_\\infty \\)</span>-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.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"31 2","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Categorical Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10485-023-09719-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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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.
期刊介绍：
Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant.
Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.
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