{"title":"Formality of $\\mathbb{E}_n$-algebras and cochains on spheres","authors":"Gijs Heuts, Markus Land","doi":"arxiv-2407.00790","DOIUrl":null,"url":null,"abstract":"We study the loop and suspension functors on the category of augmented\n$\\mathbb{E}_n$-algebras. One application is to the formality of the cochain\nalgebra of the $n$-sphere. We show that it is formal as an\n$\\mathbb{E}_n$-algebra, also with coefficients in general commutative ring\nspectra, but rarely $\\mathbb{E}_{n+1}$-formal unless the coefficients are\nrational. Along the way we show that the free functor from operads in spectra\nto monads in spectra is fully faithful on a nice subcategory of operads which\nin particular contains the stable $\\mathbb{E}_n$-operads for finite $n$. We use\nthis to interpret our results on loop and suspension functors of augmented\nalgebras in operadic terms.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.00790","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We study the loop and suspension functors on the category of augmented
$\mathbb{E}_n$-algebras. One application is to the formality of the cochain
algebra of the $n$-sphere. We show that it is formal as an
$\mathbb{E}_n$-algebra, also with coefficients in general commutative ring
spectra, but rarely $\mathbb{E}_{n+1}$-formal unless the coefficients are
rational. Along the way we show that the free functor from operads in spectra
to monads in spectra is fully faithful on a nice subcategory of operads which
in particular contains the stable $\mathbb{E}_n$-operads for finite $n$. We use
this to interpret our results on loop and suspension functors of augmented
algebras in operadic terms.