nLab internal crossed module

Idea

Crossed modules of groups, differential crossed modules (or Lie algebra crossed modules), and crossed modules of crossed modules, are a special case of an internal crossed module of objects in semi-abelian categories.

Literature

Crossed modules internal to semi-abelian categories, along with precrossed modules, were introduced in:

• George Janelidze, Internal crossed modules, Georgian Math. J. 10 1 (2003) 99–114 [doi:10.1515/GMJ.2003.99, pdf]

On further generalization to include crossed modules of monoids in even more general contexts than treated by semi-abelian categories:

• Gabriella Böhm: Crossed modules of monoids I: relative categories, Appl. Categor. Struct. 27 (2019) 641-662 [doi:10.1007/s10485-019-09570-0]

• Gabriella Böhm: Crossed modules of monoids II: Relative Crossed Modules, Appl Categor Struct 28 (2020) 601-653 [doi:10.1007/s10485-020-09592-z]

• Gabriella Böhm: Crossed modules of monoids III: Simplicial monoids of Moore length 1, Appl. Categor. Struct. 29 (2021) 1-29 [doi:10.1007/s10485-020-09603-z]

On precrossed modules in Malcev categories:

• T. Everaerta, Marino Gran, Precrossed modules and Galois theory, J. Algebra 297 1 (2006) 292-309 [doi:10.1016/j.jalgebra.2005.06.034]