an action of a Hopf algebra and with coefficients in a local system [@CM98; @hkrs1]. The local systems are closely related, and in a sense they are a mirror image of, Yetter-Drinfeld modules over a Hopf algebra. Beyond Hopf algebras, one often encounters interesting algebraic objects: for example quasi Hopf algebras and weak Hopf algebras, that only possess some of the axioms of Hopf algebras. Developing a Hopf cyclic-type theory for these Hopf-like objects is one of the motivations for this paper.

We find that the language of monoidal categories is a suitable framework to discuss this question and to discover an answer. The key point, for us, of the categorical machinery is the notion of a trace. Traces in (monoidal) categories are well-known and have been used in different settings. In [@DP] the categorical definition of ordinary traces of square matrices are used for endomorphisms of a dualizable object in a symmetric monoidal category. In the derived category of a ring, traces are called Lefschetz numbers. Categorical traces are also used to study fixed-point theory [@P]. The authors in [@JSV] showed that traces can be defined for dualizable objects in a monoidal category. On the other hand, if we 