The cotriple resolution of differential graded algebras
type: Item
instance of: scholarly article
title: The cotriple resolution of differential graded algebras
zbMATH Open document ID: 1366.18007
author: Benoît Fresse
published in: Proceedings of the American Mathematical Society
publication date: 2016-09-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1503.08489
Mathematics Subject Classification ID: 18D50
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 6629316
zbMATH Keywords: differential graded algebra
zbMATH Keywords: cotriple resolution
zbMATH Keywords: operad
zbMATH Keywords: Barrett-Eccles operad
zbMATH Keywords: \(E_\infty\)-operad
zbMATH Keywords: commutative operad
MaRDI profile type: Publication
OpenAlex ID: W3106001283
arXiv ID: 1503.08489
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DOI: 10.1090/PROC/13135
review text: In the context of algebras over operads in a homotopical setting, it is often necessary to take cofibrant resolutions. While these resolutions exist for formal reasons, it can be difficult to describe them explicitly. The purpose of this paper is to describe a cofibrant resolution of differential graded algebras over the \(E_\infty\) operad (more specifically, the Barratt-Eccles operad) and the commutative operad. The main result is that the geometric realization of the cotriple resolution of a dg-algebra \(A\) is in fact a cofibrant resolution of, and in particular weakly equivalent to, the original dg-algebra \(A\). This kind of construction has been established previously in the context of algebras in spectra, but the approaches used there cannot be applied in the dg-algebra setting.NEWLINENEWLINEAfter an initial section developing cosimplicial framings in the non-unitary dg Barratt-Eccles operad, the main result of the paper is proved in this context and then generalized to the unitary version; the proof is then adapted to the unitary commutative operad. The paper concludes with an outline of how this argument might be generalized to more general operads.
reviewed by: Julie Bergner
description: scientific article; zbMATH DE number 6629316