Strict 2-Groups are Crossed Modules

The 2-categories of strict 2-groups and crossed modules are introduced and their 2-equivalence is made explicit.

💡 Research Summary

The paper establishes a precise 2‑categorical equivalence between strict 2‑groups and crossed modules, two algebraic structures that appear in higher‑dimensional group theory and homotopy theory. A strict 2‑group is defined as an internal category in the category of groups: it consists of a group of objects (G_0) and a group of morphisms (G_1) together with source, target, identity and composition maps that are all group homomorphisms and satisfy the usual category axioms. This internal‑category viewpoint yields a 2‑category, denoted (2\text{-Grp}), whose 1‑cells are strong functors between internal categories and whose 2‑cells are natural transformations.

A crossed module, on the other hand, is a group homomorphism (\partial\colon H\to G) equipped with a left action of (G) on (H) (denoted (g\triangleright h)) satisfying the Peiffer identities:

1. (\partial(g\triangleright h)=g,\partial(h),g^{-1});
2. (\partial(h)\triangleright h’ = hh’h^{-1}). These conditions encode the interaction between the two groups in a way that mirrors the composition law of a strict 2‑group. Crossed modules form a 2‑category (X\text{Mod}) whose 1‑cells are morphisms of crossed modules (pairs of homomorphisms preserving (\partial) and the action) and whose 2‑cells are modifications between such morphisms.

The core of the paper constructs two 2‑functors \