Group completion and units in \(\mathcal I\)-spaces
type: Item
instance of: scholarly article
title: Group completion and units in \(\mathcal I\)-spaces
zbMATH Open document ID: 1277.55004
author: Steffen Sagave
author: Christian Schlichtkrull
published in: Algebraic \& Geometric Topology
publication date: 2013-05-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.6413
review text: Let \(\mathcal{I}\) be the category whose objects are the empty set and finite sets of the form \(\{1, 2, ..., n\}\), with morphisms the injective functions. Then the category \(\mathcal{S}^\mathcal{I}\) of \(\mathcal{I}\)-spaces is the symmetric monoidal category of functors \(\mathcal{I} \to \mathcal{S}\), where \(\mathcal{S}\) is the category of simplicial sets and the monoidal structure is induced by concatenation in \(\mathcal{I}\). A commutative monoid in \(\mathcal{S}^\mathcal{I}\) is known as a commutative \(\mathcal{I}\)-space monoid and, following the authors of this paper, we use \(\mathcal{C}\mathcal{S}^\mathcal{I}\) to denote the category of commutative monoids in \(\mathcal{S}^\mathcal{I}\). In [\textit{S. Sagave} and \textit{C. Schlichtkrull}, Adv. Math. 231, No. 3--4, 2116--2193 (2012; Zbl 1315.55007)], the authors of the paper under review show that \(\mathcal{C}\mathcal{S}^\mathcal{I}\) can be equipped with the ``positive \(\mathcal{I}\)-model structure,'' and the resulting model category is Quillen equivalent to the category of \(E_\infty\)-spaces. Also, if \(A \in \mathcal{C}\mathcal{S}^\mathcal{I}\), then the associated space \(A_{h\mathcal{I}}\), the homotopy colimit of the diagram \(A\), is an \(E_\infty\)-space. We now describe a result from the paper under review. Let \(B \: \mathcal{C}\mathcal{S}^\mathcal{I} \to \mathcal{C}\mathcal{S}^\mathcal{I}\) be the bar construction with right adjoint the loop functor \(\Omega\), and, with \(A\) as above, let \(B(A) \to B(A)^{\mathcal{I}-\mathrm{fib}}\) be a functorial fibrant replacement in the positive \(\mathcal{I}\)-model structure on \(\mathcal{C}\mathcal{S}^\mathcal{I}\). The authors show that if \(A\) is cofibrant in \(\mathcal{C}\mathcal{S}^\mathcal{I}\), then the natural map \[ A_{h\mathcal{I}} \to \bigl(\Omega(B(A)^{\mathcal{I}-\mathrm{fib}})\bigr)_{h\mathcal{I}} \] is a group completion. Additionally, there is a version of this result that shows that group completion for topological monoids…
reviewed by: Daniel G. Davis
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 55P43
zbMATH DE Number: 6160235
zbMATH Keywords: \(E_\infty\)-spaces
zbMATH Keywords: group completion
zbMATH Keywords: units of ring spectra
zbMATH Keywords: \(\Gamma\)-spaces
MaRDI profile type: Publication
OpenAlex ID: W2951295422
arXiv ID: 1111.6413
cites work: Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
cites work: André-Quillen cohomology of commutative \(S\)-algebras
cites work: Homology and cohomology of \(E_\infty\) ring spectra
cites work: Topological Hochschild homology of Thom spectra and the free loop space
cites work: Homotopy invariant algebraic structures on topological spaces
cites work: On the telescopic homotopy theory of spaces
cites work: Q4191283
cites work: Homotopy limits, completions and localizations
cites work: Q4716283
cites work: Simplicial homotopy theory
cites work: Q4790109
cites work: Q4220599
cites work: Spectra and symmetric spectra in general model categories
cites work: Symmetric spectra
cites work: Diagram spaces, diagram spectra and spectra of units
cites work: Q4215784
cites work: Model categories of diagram spectra
cites work: The geometry of iterated loop spaces
cites work: Q4766246
cites work: Q3182033
cites work: Diagram spaces and symmetric spectra
cites work: Units of equivariant ring spectra
cites work: Units of ring spectra and their traces in algebraic K-theory
cites work: The homotopy infinite symmetric product represents stable homotopy
cites work: Thom spectra that are symmetric spectra
cites work: Higher topological Hochschild homology of Thom spectra
cites work: S-modules and symmetric spectra
cites work: Stable homotopical algebra and Γ-spaces
cites work: Algebras and Modules in Monoidal Model Categories
cites work: A uniqueness theorem for stable homotopy theory
cites work: Categories and cohomology theories
cites work: Symmetric spectra and topological Hochschild homology
cites work: Homotopy colimits in the category of small categories
DOI: 10.2140/AGT.2013.13.625
description: scientific article