Universality of multiplicative infinite loop space machines

We establish a canonical and unique tensor product for commutative monoids and groups in an $\infty$ –category $C$ which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that $E_{n}$ –(semi)ring objects in $C$ give rise to $E_{n}$ –ring spectrum objects in $C$ . In the case that $C$ is the $\infty$ –category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic $K$ –theory of rings and ring spectra.

The main tool we use to establish these results is the theory of smashing localizations of presentable $\infty$ –categories. In particular, we identify preadditive and additive $\infty$ –categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show $Ring (D \otimes C) ≃ Ring (D) \otimes C$ . Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in $\infty$ –categories.

Keywords

infinite loop space machines, structured ring spectra, K-theory

Mathematical Subject Classification 2010

Primary: 55P48

Secondary: 55P43, 19D23

References

Publication

Received: 2 October 2013

Revised: 9 October 2014

Accepted: 28 January 2015

Published: 12 January 2016

Authors

David Gepner
Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907 USA

Moritz Groth
Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53111 Bonn Germany

Thomas Nikolaus
Mathematisches Institut der Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany

Volume 15, issue 6 (2015)

David Gepner, Moritz Groth and Thomas Nikolaus

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors