Crossed simplicial groups and their associated homology

Abstract

Abstract. 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 Braided Simplicial Group

“…It was known that the sequence of pure braid groups, K fK n 1 g n > 0 , forms a simplicial group where the i th face is given by deleting the i-string, and the i-degeneracy is given by doubling the i-string. (See, for instance, [10].) Recently Fred Cohen and the author showed that FS 1 embeds into K as a simplicial subgroup [6].…”

Zappa-Szép Products of Semigroups

“…This viewpoint on Zappa-Szép products underlies the work of Fiedorowicz and Loday [13]. In the theory of quantum groups Zappa-Szép product known as the bicrossed (bismash) product see [14].…”

The cyclic homology of crossed product algebras.

“…Shapiro's Lemma now shows that H p (G 9 A* G )^H,(G' 9 A\), [9] from which Feigin and Tsygan's form of the spectral sequence follows easily. Shapiro's Lemma now shows that H p (G 9 A* G )^H,(G' 9 A\), [9] from which Feigin and Tsygan's form of the spectral sequence follows easily.…”

“…The paracyclic category has been studied by Fiedorowicz and Loday [9] and Nistor [15]. This category Λ^, which we call \htparacyclic category has the same set of objects s the simplicial category A, namely the natural numbers n. Recall that the morphisms Δ (n, m) from n to m are the monotonically increasing maps from the set {0, ...,«} to the set {0, ...,m}.…”

“…Consider the map β from C pq (A t\ G) to C p (G, C q (A^)) given by the formula feo»···»^!^»···»^) ^ (gi>--->gp\g\<* Q9 ...,a q ) 9 where g = g 0 . Consider the map β from C pq (A t\ G) to C p (G, C q (A^)) given by the formula feo»···»^!^»···»^) ^ (gi>--->gp\g\<* Q9 ...,a q ) 9 where g = g 0 .…”