S^{2i-1}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}&=S^{2i-1}(m{~\hspace{-3pt}_{\left<2\right>}})S^{2i}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}\\
&=S^{2i-1}(S(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<2\right>}})m{~\hspace{-3pt}_{\left<0\right>}}\\
&=\varepsilon(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}\\
&=m.\end{aligned}$$

“$\Leftarrow$" $$\begin{aligned}
S^{2i}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}&=S^{2i}(m{~\hspace{-3pt}_{\left<2\right>}})S^{2i-1}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}\\
&=S^{2i-1}(m{~\hspace{-3pt}_{\left<1\right>}}S(m{~\hspace{-3pt}_{\left<2\right>}}))m{~\hspace{-3pt}_{\left<0\right>}}\\
&=\varepsilon(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}\\
&=m.\end{aligned}$$

Monoidal categories and 2-traces {#moncat2tr}
================================

This section develops the core of the conceptual machinery that we need in order to understand the Hopf-type cyclic homology theories. For convenience we start with the covariant case and derive the contravariant case from it. We note that ignoring the non-strictness of the monoidal category by suppressing the explicit formulas for associators would have cleaned up the exposition. Our choice to include them was motivated by future applications of this machinery to monoidal categories where the associator appears as an explicit formula and so would have to appear in the definitions of the cyclic structure once it is unpacked from the conceptual definitions. The ease with which such laborious formulas are safely hidden from view demonstrates the power of the categorical machinery.

Let $(\mathcal{C}, \ot)$ be a monoidal category. We will need the following conventions. Let $A$ be an object in ${\mathcal{C}}$, by $A^{\ot n}$ we mean an object defined inductively as $$A^{\ot n}=A^{\ot n-1}\ot A.$$ For $\vec{n}=(n_1,\cdots, n_k)$ with $n_i$ non-negative integers, by $A^{\ot \vec{n}}$ we denote an object defined inductively as $$A^{\ot \vec{n}}=A^{\ot (n_1,\cdots, n_{k-1})}\ot A^{\ot n_k}.$$ We interpret $A^{\ot 0}$ 