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Segal and Waldhausen K-theory: a multiplicative comparison

Banff International Research Station for Mathematical Innovation and Discovery

Fundamental work of Segal and Waldhausen gives us two versions of K-theory that produces spectra from certain types of categories. These constructions agree, in the sense that appropriately equivalent categories yield weakly equivalent spectra. In the 2000s, work of Elmendorf--Mandell and Blumberg--Mandell produced more structured versions of Segal and Waldhausen K-theory, respectively. These versions are "multiplicative," in the sense that appropriate notions of pairings of categories yield multiplication-type structure on their resulting spectra. In this talk, I will discuss joint work with Osorno in which we show that these constructions agree as multiplicative versions of K-theory. Consequently, we get comparisons of rings spectra built from these two constructions. Furthermore, the same result also allows for comparisons of related constructions of spectrally-enriched categories.

Mathematics; Category theory; homological algebra; K-theory; Algebraic topology

video/mp4

Author affiliation: Vanderbilt University

2020-09-22

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76122

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/