The simplicial interpretation of bigroupoid 2-torsors

TL;DR

This paper introduces a simplicial framework for understanding bigroupoid 2-torsors through the lens of bicategory actions, connecting categorical and simplicial perspectives in higher geometry.

Contribution

It defines an action bicategory for bicategory actions and characterizes bigroupoid 2-torsors via the Duskin nerve, linking bicategorical and simplicial structures.

Findings

Action bicategory construction for bicategory actions

Characterization of bigroupoid 2-torsors via Duskin nerve

Establishes a simplicial interpretation of bigroupoid 2-torsors

Abstract

Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, to Waldmann's work on deformation quantization. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2-functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2-torsor. In that case, the Duskin nerve of the canonical projection is precisely the Duskin-Glenn simplicial 2-torsor.