  For a (co)monad T_l on a category M, an object X in M, and a functor \Pi: M
\to C, there is a (co)simplex Z^*:=\Pi T_l^{* +1} X in C. Our aim is to find
criteria for para-(co)cyclicity of Z^*. Construction is built on a distributive
law of T_l with a second (co)monad T_r on M, a natural transformation i:\Pi T_l
\to \Pi T_r, and a morphism w: T_r X \to T_l X in M. The 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 \otimes_R (-) and T_r =
(-)\otimes_R T on the category of R-bimodules. The functor \Pi can be chosen
such that Z^n= T\hat{\otimes}_R... \hat{\otimes}_R T \hat{\otimes}_R X is the
cyclic R-module tensor product. A natural transformation i:T \hat{\otimes}_R
(-) \to (-) \hat{\otimes}_R T is given by the flip map and a morphism w: X
\otimes_R T \to T\otimes_R X is constructed whenever T is a (co)module algebra
or coring of an R-bialgebroid. Stable anti Yetter-Drinfel'd modules over
certain bialgebroids, so called x_R-Hopf algebras, are introduced. In the
particular example when T is a module coring of a x_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^{\otimes_R *+1} \otimes_B X. For a B-Galois
extension S \to T, a stable anti Yetter-Drinfel'd B-module T_S is constructed,
such that the cyclic objects B^{\otimes_R *+1} \otimes_B T_S and T^
{\hat{\otimes}_S *+1} are isomorphic. As an application, we compute Hochschild
and cyclic homology of a groupoid with coefficients, by tracing it back to the
group case. In particular, we obtain explicit expressions for ordinary
Hochschild and cyclic homology of a groupoid.
