think of a monoidal category as a bicategory with one object, then the notion of trace that we want is similar to the generalized traces of [@PS]. Finally the authors in [@FSS] found a relationship between the category-valued traces and the twisted center of a monoidal category.

The technical aspects of this paper would have been much more involved and would have required much delicacy, had the notions of bimodule categories and their centers not been already extensively studied. In particular, as we realized, the center of a certain bimodule category of the monoidal category of (left) modules over a Hopf algebra (sometimes called the twisted center of the monoidal category) provides the suitable coefficients for Hopf cyclic cohomology. These coefficients were already known, called (stable) anti Yetter-Drinfeld modules [@hkrs1], but were defined much less conceptually. We use these (or dual) central elements to form a suitable categorical trace to define the desired homological objects.

More precisely, in this paper we use the notion of a [*symmetric $2$-trace*]{} (compare with the shadow structure in [@PS]) for an abelian monoidal category to show that for a monoidal category endowed with such a trace one can attach a cyclic module to any 