
Monoidal Categories, 2Traces, and Cyclic Cohomology
In this paper we show that to a unital associative algebra object
(resp. counital coassociative coalgebra object) of any abelian
monoidal category $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$trace,
i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ satisfying some natural tracelike
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
$\mathcal{M}$ is a $\mathcal{C}$bimodule category and $(F, M)$ is a stable central
pair, i.e., $F\in Fun(\mathcal{M}, \operatorname{Vec})$ and $M\in \mathcal{M}$ satisfy certain
conditions, then $\mathcal{C}$ acquires a symmetric 2trace. 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.
Keywords:monoidal category, abelian and additive category, cyclic homology, Hopf algebra Categories:16T05, 18D10, 19D55 