The Grassmannian geometry of spectra
type: Item
instance of: scholarly article
title: The Grassmannian geometry of spectra
zbMATH Open document ID: 0681.55007
DOI: 10.1016/0022-4049(88)90023-0
author: A. D. Elmendorf
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: This paper presents a new category of spectra, \({\mathcal S}\), which the author claims has more strengths and fewer weaknesses than its rivals. The approach is inspired by Peter May's observation that spectra should be indexed by the finite-dimensional subspaces of a universe: a real inner product space topologized as the colimit of its finite dimensional subspaces. For the author a spectrum E will be a sequence of maps \(E_ k\to G_ k({\mathcal U})\) where \(G_ k({\mathcal U})\) is the space of k-planes in a universe \({\mathcal U}.\) The resulting category \({\mathcal S}\) is such that the homotopy category of CW-spectra in \({\mathcal S}\) is equivalent to Boardman's stable category. Furthermore the smash product of spectra turns \({\mathcal S}\) into a symmetric monoidal category, and for any compact group G, RO(G)-graded equivariant cohomology theories are represented by G-objects in \({\mathcal S}.\) The relationship between \({\mathcal S}\) and May's categories \({\mathcal S}{\mathcal U}\) of spectra are given as follows. The category \({\mathcal S}\) has an augmentation \(\epsilon\) : \({\mathcal S}\to {\mathcal U}n\), where \({\mathcal U}n\) is the category of universes and linear isometries. There is a canonical isomorphism of categories \({\mathcal S}{\mathcal U}\cong \epsilon^{- 1}(id_{{\mathcal U}})\).
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 4115528
zbMATH Keywords: spectra
zbMATH Keywords: Grassmannian
zbMATH Keywords: stable homotopy
zbMATH Keywords: parametrized spaces
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90023-0
OpenAlex ID: W2090273079
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description: scientific article; zbMATH DE number 4115528