3-types of simplicial groups and braided regular crossed modules
There are various algebraic models of homotopy 3-types. The authors introduce a new model: the braided regular crossed modules. They connect this to other models as follows: (i) they construct a category equivalence between the category of braided regular crossed modules and the category of simplical groups with Moore complex of length 2; (ii) they explicitly calculate a category equivalence between the category of braided regular crossed modules and the category of crossed squares; (iii) they explicitly construct a quadratic module from any braided crossed module in a functorial way. The last two items are just applications of the first one. The notions above are too involved to define here. Nevertheless, the authors recall the definition of all the models and functors they use. Their constructions are explicit and their proofs are lengthy verifications of all the requirements of the various definitions.
scientific article; zbMATH DE number 5116064