The finite stable homotopy type of some topoi
type: Item
instance of: scholarly article
title: The finite stable homotopy type of some topoi
zbMATH Open document ID: 0626.18004
DOI: 10.1016/0022-4049(87)90101-0
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let \(\epsilon\) be a Grothendieck topos and F a spectrum, in the sense of stable homotopy. The author [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)] defined the sheaf hypercohomology spectrum of \(\epsilon\), \({\mathbb{H}}(\epsilon;F)\). In this paper he constructs a spectrum, \(h\epsilon\), such that there is a natural homotopy equivalence, \((*)\quad Map(h\epsilon,F)\simeq {\mathbb{H}}(\epsilon;F)\) under suitable cohomological conditions on \(\epsilon\). When (*) holds part of its importance lies in the fact that \(h\epsilon\wedge F\) is a spectrum which represents ``the homology of \(\epsilon\) with coefficients in F''. A well-behaved homology functor on a general Grothendieck topos is unknown.
reviewed by: Victor P. Snaith
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 4017142
zbMATH Keywords: Grothendieck topos
zbMATH Keywords: spectrum
zbMATH Keywords: stable homotopy
author: Robert W. Thomason
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90101-0
OpenAlex ID: W2064797515
cites work: Q4067782
cites work: Etale homotopy
cites work: Homotopy limits, completions and localizations
cites work: Q5541099
cites work: Etale Homotopy of Simplicial Schemes. (AM-104)
cites work: Q5592097
cites work: Q5522742
cites work: Algebraic $K$-theory and etale cohomology
cites work: Comparison of equivariant algebraic and topological K-theory
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
description: scientific article; zbMATH DE number 4017142