Monoidal Categories, 2-Traces, and Cyclic Cohomology

Abstract

AbstractIn this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ($\mathscr{C},\otimes$) endowed with a symmetric 2-trace,i.e., an$F\in \text{Fun}(\mathscr{C},\text{Vec})$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$\mathscr{M}$is a$\mathscr{C}$-bimodule category and$(F,M)$is a stable central pair,i.e.,$F\in \text{Fun}(\mathscr{M},\text{Vec})$and$M\in \mathscr{M}$satisfy certain conditions, then$\mathscr{C}$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.