Universality of multiplicative infinite loop space machines

TL;DR

This paper develops a universal tensor product for commutative monoids and groups in infinity-categories, enabling a multiplicative infinite loop space machine that connects algebraic K-theory with higher algebraic structures.

Contribution

It introduces a canonical tensor product in infinity-categories, generalizes infinite loop space machines, and explores the stability of algebraic structures under basechange using smashing localizations.

Findings

Established a canonical tensor product for commutative monoids and groups in infinity-categories.

Constructed a multiplicative infinite loop space machine applicable to algebraic K-theory.

Identified preadditive and additive infinity-categories as local objects for smashing localizations.

Abstract

We establish a canonical and unique tensor product for commutative monoids and groups in an infinity-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 infinity-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 infinity-categories. In particular, we identify preadditive and additive infinity-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 infinity-categories.