Universality of multiplicative infinite loop space machines

Abstract

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 ����n –(semi)ring objects in C give rise to ����n –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(D⊗C)≃Ring(D) ⊗C). Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞–categories