[Paper Review] The simplicial interpretation of bigroupoid 2-torsors

Igor Baković|ArXiv.org|Feb 19, 2009

Homotopy and Cohomology in Algebraic Topology26 references3 citations

TL;DR

This paper establishes a simplicial interpretation of bigroupoid 2-torsors by showing that the Duskin nerve of a canonical 2-functor from a bigroupoid 2-torsor to its acting bigroupoid is precisely a Duskin-Glenn simplicial 2-torsor. The key result is that when a bigroupoid acts on a category via a principal 2-action, the resulting simplicial structure is aspherical, 1-coskeletal, and satisfies exact fibrancy conditions, thereby realizing the classical simplicial 2-torsor via categorification of groupoid 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.

Motivation & Objective

• To characterize bigroupoid 2-torsors in terms of simplicial structures.
• To establish a bridge between bicategorical actions and simplicial homotopy theory via the Duskin nerve.
• To show that principal 2-actions on categories yield simplicial 2-torsors satisfying exact fibrancy and coskeletality conditions.
• To generalize classical torsor theory to the 2-categorical setting using categorification principles.
• To provide a simplicial interpretation of 2-torsors arising from bigroupoid actions in a topos.

Proposed method

• Construct the action bicategory and a canonical strict 2-functor from the action category to the acting bigroupoid.
• Define a bigroupoid 2-torsor as a principal 2-action satisfying exact fibrancy and coskeletality conditions.
• Use the Duskin nerve to translate the 2-functor into a simplicial set.
• Prove that the nerve is aspherical and 1-coskeletal, ensuring it satisfies the conditions of a Duskin-Glenn simplicial 2-torsor.
• Leverage the fully faithful and essentially surjective nature of the induced functor on fibered products to construct unique 2-simplices.
• Verify that the nerve satisfies the exact Kan conditions above dimension 1, confirming it is a simplicial 2-torsor.

Experimental results

Research questions

• RQ1 How can bigroupoid 2-torsors be interpreted in simplicial terms using the Duskin nerve?
• RQ2 What conditions on a 2-functor ensure that its Duskin nerve is a Duskin-Glenn simplicial 2-torsor?
• RQ3 Under what conditions does a principal 2-action on a category yield a 1-coskeletal and aspherical simplicial set?
• RQ4 How does the categorification of groupoid actions relate to simplicial hypergroupoids and torsors?
• RQ5 What role does the fully faithful and essentially surjective property of the induced functor play in constructing the simplicial 2-torsor?

Key findings

• The Duskin nerve of the canonical 2-functor from a bigroupoid 2-torsor to its acting bigroupoid is a Duskin-Glenn simplicial 2-torsor.
• The nerve of the 2-torsor is aspherical, meaning it has trivial higher homotopy groups.
• The nerve is 1-coskeletal, meaning all higher simplices are determined by 1-simplices and their compositions.
• The 2-functor is an exact fibration for all n ≥ 2, ensuring the simplicial structure satisfies the required exactness conditions.
• The existence of a unique 2-simplex filling any 1-horn is guaranteed by the fully faithful and essentially surjective property of the induced functor.
• The construction establishes a precise correspondence between principal 2-actions and simplicial 2-torsors via the Duskin nerve.