Symmetric Monoidal Bicategories and Biextensions

TL;DR

This paper explores the structure of monoidal bicategories, their associated cohomological data, and how symmetry conditions relate to biextensions and MacLane cohomology, advancing the understanding of categorical extensions and their obstructions.

Contribution

It introduces a cohomological framework for analyzing monoidal bicategories and establishes connections between symmetry conditions, biextensions, and MacLane cohomology.

Findings

Monoidal structures induce biextensions of abelian groups by Picard groupoids.

Vanishing obstructions correspond to symmetry conditions on biextensions.

Computations relate to the cubical Q-construction and MacLane (co)homology.

Abstract

We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy of possible commutativity conditions in terms of progressive stabilization of these data. We also show that monoidal structures on bicategories give rise to biextensions of a pair of (abelian) groups by a Picard groupoid, and that the progressive vanishing of obstructions determined by the tower of commutative structures corresponds to appropriate symmetry conditions on these biextensions. In the fully symmetric case, which leads us fully into the stable range, we show how our computations can be expressed in terms of the cubical Q-construction underlying MacLane (co)homology.