Stable Homotopy Seminar, 13: The Smash Product
Channel: Paul VanKoughnett
Date: 11/05/2021

I describe a bunch of desirable properties of the smash product on spectra, and then prove Lewis's theorem that no category of spectra has them. However, there are ways of sacrificing one or the other of the properties and getting something fairly well-behaved. The earliest attempts are Boardman's and Adams's "hand-crafted smash products", which define a symmetric monoidal structure on the stable homotopy category but not on Spectra itself, and require a lot of technical manipulations to work right. We then look at the symmetric spectra of Hovey-Shipley-Smith and the orthogonal spectra of Lewis-May, which are actual symmetric monoidal model categories. Both fit into the Mandell-May-Schwede-Shipley formalism of "diagram spectra", in which the natural numbers which index spectra are replaced by a richer symmetric monoidal indexing category -- finite sets in the case of symmetric spectra, finite-dimensional real inner product spaces in the case of orthogonal spectra.


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This is part of the stable homotopy learning seminar organized by Paul VanKoughnett. For notes and more, visit https://www.math.purdue.edu/~pvankoug/spectra/index.html