[Paper Review] The inner automorphism 3-group of a strict 2-group

David Roberts, Urs Schreiber|arXiv (Cornell University)|Aug 13, 2007

Homotopy and Cohomology in Algebraic Topology14 references21 citations

TL;DR

This paper introduces the inner automorphism 3-group INN₀(G(2)) for any strict 2-group G(2), showing it fits into a short exact sequence G(2) → INN₀(G(2)) → BG(2). The construction arises from the mapping cone of the identity on G(2), and INN₀(G(2)) is shown to be contractible and universal for G(2)-2-bundles, generalizing the classical universal G-bundle via INN(G).

ABSTRACT

Any group $G$ gives rise to a 2-group of inner automorphisms, $\mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $\mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $\mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $\mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} o \mathrm{INN}_0(G_{(2)}) o ΣG_{(2)}$. As a consequence, $\mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.

Motivation & Objective

• To define and characterize the inner automorphism 3-group INN₀(G(2)) for any strict 2-group G(2).
• To establish that INN₀(G(2)) fits into a short exact sequence G(2) → INN₀(G(2)) → BG(2) of 2-groupoids.
• To show that INN₀(G(2)) is contractible and universally encodes the structure of the G(2)-2-bundle.
• To relate INN₀(G(2)) to the mapping cone of the identity on G(2), providing a homotopical interpretation.
• To generalize the classical universal G-bundle construction via INN(G) to higher categorical bundles using 3-groups.

Proposed method

• Construct INN₀(G(2)) as a 3-group arising from the mapping cone of the identity crossed module of G(2).
• Use the double nerve construction to realize G(2) as a bisimplicial set NG(2), then apply the décalage functor Dec₁ to obtain a contractible simplicial set.
• Show that the resulting nerve of INN₀(G(2)) is isomorphic to Dec₁NG(2), which fits into an exact sequence with the fiber N'G(2) and base BG(2).
• Demonstrate that the 2-groupoid underlying INN₀(G(2)) is equivalent to the tangent 2-category of BG(2), via the décalage construction.
• Use the structure of 2-crossed modules to describe the 3-group INN₀(G(2)) explicitly in terms of objects, morphisms, and 2-morphisms.
• Prove that INN₀(G(2)) is contractible by showing its nerve is weakly contractible via the décalage and realization of bisimplicial sets.

Experimental results

Research questions

• RQ1 How can the notion of inner automorphisms be generalized from groups to 2-groups?
• RQ2 What is the structure of the inner automorphism 3-group INN₀(G(2)) for a strict 2-group G(2)?
• RQ3 How does INN₀(G(2)) relate to the universal G(2)-2-bundle and the classifying 2-groupoid BG(2)?
• RQ4 Can the construction of INN₀(G(2)) be interpreted as a mapping cone of the identity on G(2)?
• RQ5 What is the homotopical significance of INN₀(G(2)), particularly in terms of contractibility and universality?

Key findings

• The inner automorphism 3-group INN₀(G(2)) is constructed as a 3-group whose underlying 2-groupoid fits into a short exact sequence G(2) → INN₀(G(2)) → BG(2).
• INN₀(G(2)) is contractible as a 2-groupoid, meaning it is equivalent to the trivial 2-group, which implies its nerve is weakly contractible.
• The 3-group INN₀(G(2)) arises naturally as the nerve of the décalage of the double nerve of G(2), i.e., N(INN₀(G(2))) ≅ Dec₁NG(2).
• The construction of INN₀(G(2)) is equivalent to the mapping cone of the identity crossed module of G(2), providing a homotopical interpretation.
• The realization |N(INN₀(G(2)))| is contractible, and |N(INN₀(G(2)))| ≃ |NG(2)| = BG(2), confirming its role as a universal 2-bundle.
• The 2-crossed module structure of INN₀(G(2)) is explicitly described, with the crossed square arising from the identity on the crossed module H → G.