Crossed simplicial groups and their associated homology

  title={Crossed simplicial groups and their associated homology},
  author={Zbigniew Fiedorowicz and Jean-Louis Loday},
  journal={Transactions of the American Mathematical Society},
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… 
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary
Cyclic Spaces and S 1-Equivariant Homology
There are several ways of constructing simplicial models for the circle S 1. The simplest one consists in taking only two non-degenerate cells: one in dimension 0 and one in dimension 1. Another
An Introduction to Hochschild and Cyclic Homology
We define Hochschild and cyclic (co)homology for simplicial and cyclic modules. The theory for an algebra A is then obtained from the canonical simplicial/cyclic module C∗(A) associated to this
Homological and homotopical constructions for functors on ordered groupoids
The main topic of this thesis is the generalization to ordered groupoids of some results and constructions that have arisen in groupoid theory and its applications in homological and homotopical
The cyclic homology of crossed product algebras. II. Topological algebras
This article is a sequel to [6], in which we constructed a spectral sequence for the cyclic homology of a crossed product algebra A o G, where A is an algebra and G is a discrete group acting on A.
The cyclic homology of crossed product algebras
In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic
Filtration of cohomology via symmetric semisimplicial spaces
Inspired by Deligne’s use of the simplicial theory of hypercoverings in defining mixed Hodge structures ([Del75]), we replace the indexing category ∆ by the symmetric simplicial category ∆S and study
On the geometric realization and subdivisions of dihedral sets
By expressing the geometric realization of simplicial sets and cyclic sets as filtered colimits, Drinfeld (arXiv:math/0304064v3) proved in a substantially simplified way the fundamental facts that
From manifolds to invariants of En-algebras
This thesis is the first step in an investigation of an interesting class of invariants of En-algebras which generalize topological Hochschild homology. The main goal of this thesis is to simply give
Representations for certain crossed simplicial groups generated by braided Hopf algebras
We find solutions of a nonlinear equation which provide representations for the new groups R(n) defined in [1]. The groups are crossed simplicial groups in the sense of Loday [8] (Sec.6.3). These


Discrete subgroups of Lie groups
Preliminaries.- I. Generalities on Lattices.- II. Lattices in Nilpotent Lie Groups.- III. Lattices in Solvable Lie Groups.- IV. Polycyclic Groups and Arithmeticity of Lattices in Solvable Lie
Homotopy Limits, Completions and Localizations
Completions and localizations.- The R-completion of a space.- Fibre lemmas.- Tower lemmas.- An R-completion of groups and its relation to the R-completion of spaces.- R-localizations of nilpotent
Categories for the Working Mathematician
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large
Homologies diédrale et quaternionique, Adv. in Math
  • Homologies diédrale et quaternionique, Adv. in Math
  • 1987
Loday, Homologies diédrale et quaternionique
  • Adv. in Math
  • 1987
Groupes simpliciaux croisés symétriques et hyperoctahedral
  • Groupes simpliciaux croisés symétriques et hyperoctahedral
  • 1986
Homologie cyclique: produits
  • généralisations, Preprint, IRMA, Strasbourg
  • 1986