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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.15900 |
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Table of Contents:
- This is an overview of the idea of a crossed module. For a group, the triple that consists of the group, its group of automorphisms, and the canonical homomorphism from the group to its group of automorphisms constitutes a crossed module. Crossed modules arise from the identities among the relations of the presentation of a group, from the extension problem for groups and, more generally, in low dimensional topology. Also, the (successful) attempt to extend the idea of a normal extension of commutative fields to the realm of non-commutative algebras leads to crossed modules. Crossed modules appear implicitly in a forgotten paper by A. Turing which in principle settles the extension problem for groups. Crossed modules make perfect sense for Lie algebras.