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A note on Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

In our previous work, Hopf cyclic cohomology in braided monoidal categories, we extended the formalism of Hopf cyclic cohomology due to Connes and Moscovici and the more general case of Hopf cyclic cohomology with coefficients to the context of abelian braided monoidal categories. In this paper we go one step further in reducing the restriction of the ambient category being symmetric. We let the ambient category to be non-symmetric but assume only the restriction on the braid map for the Hopf algebra object (in that category) which is the main player in the theory. In the case of Hopf cyclic cohomology with (nontrivial) coefficients we also need to have similar restrictions on the braid map for the object(s) providing the coefficients datum. We present a family of examples of non-symmetric categories in which many objects with such a restrictions on the braid map exist (anyonic vector spaces).