Notes on weak units of group-like 1- and 2-stacks

Abstract

AbstractThe weak units of strict monoidal 1- and 2-categories are defined respectively in [15] and [14]. 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. We consider the (2-)substack of cancelable objects of a monoidal 1-(2-)stack. We observe that this (2-)substack is trivially group-like, its weak units are the same as the weak units of the monoidal 1-(2-)stack, and therefore we can recover the contractibility results in [15] and [14] by analysing it.