Crossed module
From Groupprops
Definition
Suppose and are groups. A crossed module structure for over (i.e., with being the crossed module over ) is the following data:
- A homomorphism of groups
- A group action of on , i.e., a homomorphism of groups
satisfying the following conditions:
- The map pushes forward via to the conjugation action of on itself:
- The map pulls back via to the conjugation action of on itself:
The conditions are more easily stated if we use to denote all the conjugation actions within a group and to denote the action , so that becomes . In that case, the conditions become:
Related notions
Facts
References
- Combinatorial Homotopy II. by J. H. C. Whitehead, Volume 55, Page 453 - 496(Year 1949): ^{Official PDF (ungated)}^{More info}
Other uses
- Van Kampen theorems for diagrams of spaces by Ronald Brown and Jean-Louis Loday, Topology, Volume 26,Number 3, Page 311 - 335(Year 1987): ^{ungated copy (PDF)}^{More info}, Section 1 beginning