In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category (
) endowed with a symmetric 2-trace, i.e., an
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
”. Furthermore, we observe that if
-bimodule category and
is a stable central pair, i.e.,
satisfy certain conditions, then
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.