Volume 15, issue 6 (2015)

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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
Abstract

We establish a canonical and unique tensor product for commutative monoids and groups in an $\infty$–category $\mathsc{C}$ which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that ${\mathbb{E}}_{n}$–(semi)ring objects in $\mathsc{C}$ give rise to ${\mathbb{E}}_{n}$–ring spectrum objects in $\mathsc{C}$. In the case that $\mathsc{C}$ is the $\infty$–category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic $K\phantom{\rule{0.3em}{0ex}}$–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 $Ring\left(\mathsc{D}\otimes \mathsc{C}\right)\simeq Ring\left(\mathsc{D}\right)\otimes \mathsc{C}$. 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