Abstract
For a (co)monad T l on a category \({\mathcal{M}}\), an object X in \({\mathcal{M}}\) , and a functor \({{\varvec {\Pi}}:\mathcal{M} \to \mathcal{C}}\) , there is a (co)simplex \({Z^\ast:={\varvec {\Pi} {T_l}}^{\ast +1} X}\) in \({\mathcal{C}}\) . 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 \({\mathcal{M}}\) , a natural transformation \({i:{\varvec {\Pi} {T_l} \to {\bf \Pi} {T_r}}}\) , and a morphism \({w:{\varvec {T_r}}X \to {\varvec {T_l}}X}\) in \({\mathcal{M}}\) . 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 \({{\varvec {T_l}}=T \otimes_R (-)}\) and \({{\varvec {T_r}}=(-)\otimes_R T}\) on the category of R-bimodules. The functor Π can be chosen such that \({Z^n=T\widehat{\otimes}_R\cdots \widehat{\otimes}_R T \widehat{\otimes}_RX}\) is the cyclic R-module tensor product. A natural transformation \({{i}:T \widehat{\otimes}_R (-) \to (-) \widehat{\otimes}_R T}\) 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. 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 \({\mathcal{B}}\) and X is a stable anti-Yetter-Drinfel’d \({\mathcal{B}}\) -module, the para-cyclic object Z * is shown to project to a cyclic structure on \({T^{\otimes_R\, \ast+1} \otimes_{\mathcal{B}} X}\) . For a \({\mathcal{B}}\) -Galois extension \({S \subseteq T}\) , a stable anti-Yetter-Drinfel’d \({\mathcal{B}}\) -module T S is constructed, such that the cyclic objects \({\mathcal{B}^{\otimes_R\, \ast+1} \otimes_{\mathcal{B}} T_S}\) and \({T^{\widehat{\otimes}_S\, \ast+1}}\) 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.
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Böhm, G., Ştefan, D. (Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads. Commun. Math. Phys. 282, 239–286 (2008). https://doi.org/10.1007/s00220-008-0540-3
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DOI: https://doi.org/10.1007/s00220-008-0540-3