### Abstract

For a (co)monad T _{l} on a category M, an object X in M, and a functor Π : M → C, there is a (co)simplex Z^{z.ast;}:= Π T_{l} ^{*}+1 X in C. The aim of this paper is to find criteria for para-(co)cyclicity of Z *. Our construction is built on a distributive law of T _{l} with a second (co)monad T _{r} on M, a natural transformation i:Π T_{l}} to Π} T_{r}, and a morphism {w:{T_{r}}}X to T_{l}}X in M. The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun's axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads T _{l}}=T ⊗_{R} (-) and T_{r}}}=(-) ⊗_{R} T} on the category of R-bimodules. The functor Π can be chosen such that {Z^{n}=T ⊗ _{R} ⊗ _{R} T ⊗_{RX}} is the cyclic R-module tensor product. A natural transformation {i}:T ⊗_{R} (-) \to (-) \widehat⊗ _{R} T} is given by the flip map and a morphism {w: X ⊗_{R} T to T⊗ R X} is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel'd module over certain bialgebroids, the so-called × _{R} -Hopf algebras, is introduced. In the particular example when T is a module coring of a × _{R} -Hopf algebra B and X is a stable anti-Yetter-Drinfel'd B-module, the para-cyclic object Z _{*} is shown to project to a cyclic structure on T^{⊗} _{R}, *+1 ⊗_{B}X. For a B-Galois extension ⊂ T}, a stable anti-Yetter-Drinfel'd B-module T _{S} is constructed, such that the cyclic objects B^{⊗} _{R}., *+1 ⊗_{B} T_{S} and {T^{⊗} _{S} *+1 are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel'd module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.

Original language | English |
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Pages (from-to) | 239-286 |

Number of pages | 48 |

Journal | Communications in Mathematical Physics |

Volume | 282 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 2008 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*282*(1), 239-286. https://doi.org/10.1007/s00220-008-0540-3