Monoidal Categories, 2-Traces, and Cyclic Cohomology

Abstract

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category C endowed with a symmetric 2-trace, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the cyclic (co)homology of the (co)algebra "with coefficients in F ". We observe that if M is a C-bimodule category equipped with a stable central pair then C acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories, obtain a conceptual understanding of anti-Yetter-Drinfeld modules, and give a formula-free definition of cyclic cohomology. The machinery can also be applied in settings more general than Hopf algebra modules and comodules.2010 Mathematics Subject Classification. monoidal category (18D10), abelian and additive category (18E05), cyclic homology (19D55), Hopf algebras (16T05).

Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

“…Remark 2.7. The current approach to general stability of aYD modules, via stability of aYD contramodules (implicitly as in [9] or explicitly as in Definition 2.5), may seem unsatisfactory but it is the only way in general. In particular situations one may do better.…”

“…The resulting categories are isomorphic to each other, and in turn isomorphic to the center of a certain bimodule category. As in [9] consider a monoidal functor # : H M → H M, taking a left H-module M to M # , where M # is the same as M as a k-vector space but the module structure is modified by S 2…”

“…If H is a Hopf algebra, it was proved in [9] that the center Z H M ( # H M) is the same as anti-Yetter-Drinfeld modules. We are going to use this fact as a guide and give a description of aYD modules in the quasi-Hopf case.…”

A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

“…An approach to this problem is afforded by [12] where it is shown that the monoidalcategorical point of view provides a way to extract the necessary formulas from a conceptual model of the situation. It is demonstrated that indeed one obtains the usual definitions in the Hopf algebra case.…”

“…The paper is organized as follows. In Section 2 we discuss some generalities as in [12,22] concerning a conceptual definition of contramodule coefficients for a suitable monoidal category. A new phenomenon that arises in this paper is the difficulty of proving that a certain natural map is an isomorphism.…”