---
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 ${\mathcal{C}}$ endowed with a symmetric $2$-trace, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the cyclic (co)homology of the (co)algebra “with coefficients in $F$". We observe that if ${\mathcal{M}}$ is a ${\mathcal{C}}$-bimodule category equipped with a stable central pair 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, obtain a conceptual understanding of anti-Yetter-Drinfeld modules, and give a formula-free definition of cyclic cohomology. The machinery can also be applied in settings more general than Hopf algebra modules and comodules.'
author:
- 'Mohammad Hassanzadeh, Masoud Khalkhali, Ilya Shapiro'
title: 'Monoidal Categories, 2-Traces, and Cyclic Cohomology'
---

monoidal category (18D10), abelian and additive category (18E05), cyclic homology (19D55), Hopf algebras (16T05).

 Introduction
=============

One of the major advances in cyclic cohomology theory in recent years was the introduction of a new cohomology theory for Hopf algebras by Connes and Moscovici and its extension to a cohomology theory for (co)algebras endowed with 