If a Bousfield localization of spectra $L_E$ at a spectrum $E$ preserves all direct sums, then it is given by smash product with the $E$-localization of the sphere spectrum
and is hence called a smashing localization.
Smashing localizations hence in particular
preserve (∞,1)-colimits;
are monoidal (∞,1)-functors except possibly for preservation of the tensor unit.
see e.g. (GGN 13, p. 8) for discussion.
rationalization is smashing: $L_{\mathbb{Q}} \simeq (-) \wedge H \mathbb{Q}$ (e.g Bauer 11, example 1.7 (4))
For $G$ a group without torsion, then localization at the Moore spectrum $E = S G$ (in particular p-localization) is smashing (see at Bousfield localization of spectra here).
Localization with respect to Morava E-theory is smashing (Hopkins-Ravenel).
“finite localizations” are smashing (Miller 92)
Jacob Lurie, Chromatic Homotopy Theory Lecture notes, Lecture 22 Morava E-theory and Morava K-theory (pdf)
Miller, Finite localizations, Boletin de la Sociedad Matematica Mexicana 37 (1992), 383–390 (HopfArchive)
David Gepner, Moritz Groth, Thomas Nikolaus, Universality of multiplicative infinite loop space machines (arXiv:1305.4550)
Tilman Bauer, Bousfield localization and the Hasse square (2011) (pdf)
Denis Nardin, section 3.2 of Stability and distributivity over orbital ∞-categories, 2012 (pdf)
Last revised on September 20, 2018 at 12:25:18. See the history of this page for a list of all contributions to it.