Crossed simplicial groups and their associated homology

@article{Fiedorowicz1991CrossedSG,
  title={Crossed simplicial groups and their associated homology},
  author={Zbigniew Fiedorowicz and Jean-Louis Loday},
  journal={Transactions of the American Mathematical Society},
  year={1991},
  volume={326},
  pages={57-87}
}
We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes' results can be generalized and simplified in this framework. A simplicial set (resp. group) is a family of sets (resp. groups) {Gn}n>0 together with maps (resp. group homomorphisms) which satisfy some well-known universal formulas. The… 
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