Notes on Weak Units of Group-Like 1- and 2-Stacks

TL;DR

This paper extends the concept of weak units from strict monoidal categories to group-like 1- and 2-stacks, demonstrating their structure as contractible Picard stacks and providing a cohomological framework.

Contribution

It introduces the definition of weak units for group-like 1- and 2-stacks and describes their structure as contractible Picard stacks with a cohomological representation.

Findings

Weak units form contractible Picard 1- and 2-stacks.

Cohomological description via complexes of sheaves of groups.

Extension to the monoidal case.

Abstract

The weak units of strict monoidal 1- and 2-categories are already defined. In this paper, we define them for group-like 1- and 2-stacks. We show that they form a contractible Picard 1- and 2-stack, respectively. We give their cohomological description which provides for these stacks a representation by complexes of sheaves of groups. Later, we extend the discussion to the monoidal case.