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Universality of
multiplicative infinite loop space machines
David Gepner, Moritz Groth and Thomas Nikolaus |
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Algebraic & Geometric Topology 15 (2015) 3107–3153
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Abstract |
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We establish a canonical and unique tensor product for commutative monoids and groups in an –category which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that –(semi)ring objects in give rise to –ring spectrum objects in . In the case that is the –category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic –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 . Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in –categories. |
Keywords
infinite loop space machines, structured ring spectra,
K-theory
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Mathematical Subject Classification 2010
Primary: 55P48
Secondary: 55P43, 19D23
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References |
Publication
Received: 2 October 2013
Revised: 9 October 2014
Accepted: 28 January 2015
Published: 12 January 2016
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Authors |
| David Gepner | |
| Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907 USA |
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| Moritz Groth | |
| Max-Planck-Institut für
Mathematik Vivatsgasse 7 D-53111 Bonn Germany |
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| Thomas Nikolaus | |
| Mathematisches Institut der
Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany |
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