Infinite loop spaces, and coherence for symmetric monoidal bicategories

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.

31 pages

References in corpus (4)

• Constructing symmetric monoidal bicategories
• Biequivalences in tricategories
• Cartesian Bicategories II
• Modeling Stable One-Types

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