On weak maps between 2-groups

TL;DR

This paper introduces butterflies as a simple, cocycle-free way to describe weak maps between 2-groups, establishing a bicategory equivalent to homotopy 2-types and clarifying key 2-group concepts.

Contribution

It provides a new explicit framework for weak maps between 2-groups using butterflies, simplifying the understanding of their composition and related structures.

Findings

Defined butterflies as a new tool for 2-group morphisms

Established a bicategory biequivalent to homotopy 2-types

Simplified descriptions of kernels, cokernels, and extensions in 2-group theory

Abstract

We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2-category of pointed homotopy 2-types. We indicate how certain standard notions of 2-group theory (e.g., kernels, cokernels, extension of 2-groups, and so on) find a simple description in terms of butterflies. We also discuss braided and abelian butterflies.