Infinite loop spaces, and coherence for symmetric monoidal bicategories

TL;DR

This paper establishes coherence theorems for symmetric monoidal bicategories, demonstrating their equivalence to strict models and exploring topological applications like E-infinity structures and monoidal properties of fundamental 2-groupoids.

Contribution

It proves three coherence theorems for symmetric monoidal bicategories and applies these results to topological structures such as classifying spaces and fundamental 2-groupoids.

Findings

Diagrams of 2-cells in free symmetric monoidal bicategories commute

The free symmetric monoidal bicategory on one object is equivalent to a disjoint union of symmetric groups

Every symmetric monoidal bicategory is equivalent to a strict one

Abstract

This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one. We give two topological applications of these coherence results. First, we show that the classifying space of a symmetric monoidal bicategory can be equipped with an E_{\infty} structure. Second, we show that the fundamental 2-groupoid of an E_n space, n \geq 4, has a symmetric monoidal structure. These calculations also show that the fundamental 2-groupoid of an E_3 space has a sylleptic monoidal structure.