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Universality of multiplicative infinite loop space machines

David Gepner, Moritz Groth and Thomas Nikolaus

Algebraic & Geometric Topology 15 (2015) 3107–3153

We establish a canonical and unique tensor product for commutative monoids and groups in an –category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En–(semi)ring objects in C give rise to En–ring spectrum objects in C. In the case that C is the –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 –categories. In particular, we identify preadditive and additive –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(DC) Ring(D) C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in –categories.

infinite loop space machines, structured ring spectra, K-theory
Mathematical Subject Classification 2010
Primary: 55P48
Secondary: 55P43, 19D23
Received: 2 October 2013
Revised: 9 October 2014
Accepted: 28 January 2015
Published: 12 January 2016
David Gepner
Department of Mathematics
Purdue University
150 N. University Street
West Lafayette, IN 47907
Moritz Groth
Max-Planck-Institut für Mathematik
Vivatsgasse 7
D-53111 Bonn
Thomas Nikolaus
Mathematisches Institut der Universität Bonn
Endenicher Allee 60
D-53115 Bonn