Tensor product of operads and iterated loop spaces
type: Item
instance of: scholarly article
title: Tensor product of operads and iterated loop spaces
zbMATH Open document ID: 0672.55004
DOI: 10.1016/0022-4049(88)90103-X
author: Gerald Dunn
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: We use the terminology of \textit{J. P. May} [The geometry of iterated loop spaces, Lect. Notes Math. 271 (1972; Zbl 0244.55009)]. Let \({\mathcal C}_ n\) denote the operad of little n-cubes, which describe n-fold loop structures on a topological space. The n canonical inclusions \({\mathcal C}_ 1\to {\mathcal C}_ n\) define an inclusion of the n-fold tensor product \({\mathcal C}_ 1^{\otimes n}\) of \({\mathcal C}_ 1\) in \({\mathcal C}_ n\). The main result of the present paper is that this inclusion is a local \(\Sigma\)-equivalence. Hence any \({\mathcal C}_ n\)-space is (weakly) homotopy equivalent to a \({\mathcal C}_ 1^{\otimes n}\)-space. This, in particular, implies that an n-fold loop structure is up to homotopy the same as n commuting one-fold loop structures. The author also extends the passage from loop structures to Moore loop structures (which are strictly associative) to \({\mathcal C}\)-spaces for an arbitrary operad \({\mathcal C}\) and analyses what happens to a \({\mathcal C}^{\otimes n}\)-space after this passage: It becomes a space with n strict monoid structures which commute with each other on suitable explicitly given subsets, but not globally (recall, two commuting monoid structures define the structure of an abelian monoid).
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 4100171
zbMATH Keywords: iterated loop spaces
zbMATH Keywords: space with monoid structure
zbMATH Keywords: operad of little n- cubes
zbMATH Keywords: n-fold loop structures
zbMATH Keywords: commuting one-fold loop structures
zbMATH Keywords: Moore loop structures
zbMATH Keywords: operad
reviewed by: Rainer Vogt
MaRDI profile type: Publication
cites work: Infinite Loop Spaces (AM-90)
cites work: Homotopy-everything 𝐻-spaces
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cites work: Uniqueness of \(n\)-fold delooping machines
cites work: Classifying Spaces of Topological Monoids and Categories
cites work: Homology fibrations and the 'group-completion' theorem
cites work: Q3950708
cites work: The geometry of iterated loop spaces
cites work: Q4766246
cites work: Pairings of categories and spectra
Wikidata QID: Q127720134
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description: scientific article; zbMATH DE number 4100171