Stable Homotopy Theory of Simplicial Presheaves

1987 ◽  
Vol 39 (3) ◽  
pp. 733-747 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Model ◽  
Stable Homotopy Theory ◽  
Simplicial Sets ◽  
Stable Homotopy Category ◽  
Closed Model Category ◽  
Simplicial Presheaves ◽  
Direct Use

Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre(C) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre(C)stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [1], and makes direct use of their Theorem A.7.This result gives a precise description of the associated stable homotopy category Ho(S Pre(C))stab, according to well known results of Quillen [6]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.

scholarly journals Motivic and real étale stable homotopy theory

2018 ◽  
Vol 154 (5) ◽  
pp. 883-917 ◽  
Author(s):  
Tom Bachmann
Keyword(s):  
Homotopy Theory ◽  
Finite Dimension ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
The Real ◽  
Rigidity Results ◽  
Noetherian Scheme ◽  
Stable Homotopy Category

Let$S$be a Noetherian scheme of finite dimension and denote by$\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$the (additive inverse of the) morphism corresponding to$-1\in {\mathcal{O}}^{\times }(S)$. Here$\mathbf{SH}(S)$denotes the motivic stable homotopy category. We show that the category obtained by inverting$\unicode[STIX]{x1D70C}$in$\mathbf{SH}(S)$is canonically equivalent to the (simplicial) local stable homotopy category of the site$S_{\text{r}\acute{\text{e}}\text{t}}$, by which we mean thesmallreal étale site of$S$, comprised of étale schemes over$S$with the real étale topology. One immediate application is that$\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the$\unicode[STIX]{x1D70C}$-local sphere (over$\mathbb{R}$). As further applications we show that$D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$(improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that$\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$for$i=1,2$and establish some new rigidity results.

scholarly journals Stable homotopical algebra and Γ-spaces

1999 ◽  
Vol 126 (2) ◽  
pp. 329-356 ◽  
Author(s):  
STEFAN SCHWEDE
Keyword(s):  
Homotopy Theory ◽  
Model Category ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Develop Model ◽  
Smash Product ◽  
Spectral Sequences ◽  
Homotopical Algebra ◽  
Stable Homotopy Theory ◽  
Set Up

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

scholarly journals Stability in pro-homotopy theory

1990 ◽  
Vol 33 (3) ◽  
pp. 419-441
Author(s):  
R. M. Seymour
Keyword(s):  
Stability Problem ◽  
Homotopy Theory ◽  
Category Structure ◽  
Local Condition ◽  
Model Category ◽  
Homotopy Category ◽  
Suitable Model ◽  
Closed Model ◽  
Closed Model Category ◽  
The Stability

If is a category, an object of pro- is stable if it is isomorphic in pro- to an object of . A local condition on such a pro-object, called strong-movability, is defined, and it is shown in various contexts that this condition is equivalent to stability. Also considered, in the case is a suitable model category, is the stability problem in the homotopy category Ho(pro-), where pro- has the induced closed model category structure defined by Edwards and Hastings [6].

scholarly journals Remarks on motivic homotopy theory over algebraically closed fields

2010 ◽  
Vol 7 (1) ◽  
pp. 55-89 ◽  
Author(s):  
Po Hu ◽  
Igor Kriz ◽  
Kyle Ormsby
Keyword(s):  
Homotopy Theory ◽  
Homotopy Category ◽  
Stable Homotopy ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Motivic Homotopy Theory ◽  
Stable Homotopy Category ◽  
Motivic Homotopy ◽  
Algebraically Closed Fields

AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

v 1 - and v 2 -Periodicity in Stable Homotopy Theory

1981 ◽  
Vol 103 (4) ◽  
pp. 615 ◽  
Author(s):  
Donald M. Davis ◽  
Mark Mahowald
Keyword(s):  
Homotopy Theory ◽  
Stable Homotopy ◽  
Stable Homotopy Theory

Spectra of derived module homomorphisms

1987 ◽  
Vol 101 (2) ◽  
pp. 249-257 ◽  
Author(s):  
Alan Robinson
Keyword(s):  
Spectral Sequence ◽  
Homotopy Theory ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Theory ◽  
Ring Spectrum ◽  
Homotopy Invariant ◽  
New Construction

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

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