Ring operads and symmetric bimonoidal categories

TL;DR

This paper introduces a new framework called ring operad theory to model $E_infty$ ring spaces, connecting classical operad pair theory with symmetric bimonoidal categories and providing alternative constructions and comparisons.

Contribution

It generalizes classical operad pair theory to a new model for $E_infty$ ring spaces, linking symmetric bimonoidal categories with $E_infty$ ring spaces.

Findings

Classifying spaces of symmetric bimonoidal categories are homeomorphic to certain $E_infty$ ring spaces.

Provides an alternative construction from symmetric bimonoidal categories to $E_infty$ ring spaces.

Establishes a connection allowing classical multiplicative infinite loop space machines to be applied to $E_infty$ ring operad algebras.

Abstract

We generalize the classical operad pair theory to a new model for $E_\infty$ ring spaces, which we call ring operad theory, and establish a connection with the classical operad pair theory, allowing the classical multiplicative infinite loop machine to be applied to algebras over any $E_\infty$ ring operad. As an application, we show that classifying spaces of symmetric bimonoidal categories are directly homeomorphic to certain $E_\infty$ ring spaces in the ring operad sense. Consequently, this provides an alternative construction from symmetric bimonoidal categories to classical $E_\infty$ ring spaces. We also present a comparison between this construction and the classical approach.