Monoidal Categories, 2-traces, and Cyclic Cohomology

In this paper we show that to a unital associative algebra object (resp. co-unital coassociative co-algebra object) of any abelian monoidal category (C,⊗) endowed with a symmetric 2-trace, i.e. an F ∈ Fun(C,Vec) satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in F”. Furthermore, we observe that if M is a C-bimodule category and (F,m) is a stable central pair, i.e., F ∈ Fun(M,Vec) and m ∈ M satisfy certain conditions, 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. 2010 Mathematics Subject Classification. monoidal category (18D10), abelian and additive category (18E05), cyclic homology (19D55), Hopf algebras (16T05).