May's Conjecture on Bimonoidal Functors and Multiplicative Infinite Loop Space Theory

TL;DR

This paper proves a weaker version of May's conjecture, showing that certain symmetric bimonoidal functors can be strictified, which simplifies the construction of multiplicative infinite loop space machines in algebraic topology.

Contribution

It establishes that multiplicatively strong symmetric bimonoidal functors can be replaced by strict ones in the context of May's infinite loop space theory.

Findings

Weaker form of May's conjecture proved.

Strictification of bimonoidal functors achieved.

Application to infinite loop space machines demonstrated.

Abstract

A conjecture of May states that there is an up-to-adjunction strictification of symmetric bimonoidal functors between bipermutative categories. The main result of this paper proves a weaker form of May's conjecture that starts with multiplicatively strong symmetric bimonoidal functors. As the main application, for May's multiplicative infinite loop space machine from bipermutative categories to either E-infinity ring spaces or E-infinity ring spectra, multiplicatively strong symmetric bimonoidal functors can be replaced by strict symmetric bimonoidal functors.