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# Monoidal Categories, 2-Traces, and Cyclic Cohomology

Published:2018-05-18

Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4
• Masoud Khalkhali,
Department of Mathematics, University of Western Ontario, London N6A 5B7, Ontario N6A 5B7
• Ilya Shapiro,
Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4
 Format: LaTeX MathJax

## Abstract

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 $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$-trace, i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ 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 $\mathcal{M}$ 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 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.
 Keywords: monoidal category, abelian and additive category, cyclic homology, Hopf algebra
 MSC Classifications: 16T05 - Hopf algebras and their applications [See also 16S40, 57T05] 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 19D55 - $K$-theory and homology; cyclic homology and cohomology [See also 18G60]

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