F(-\ot c)  \rightarrow F(c\ot -).$$

-   A 2-trace $F$ is called a symmetric 2-trace (compare with the shadow structure in [@PS]) if $$\iota_{c}(1)=F({\mathfrak{a}}_{0,1}^{1,0}).$$

Note that the symmetry condition is indeed worthy of its name as it ensures that $$\iota_c(c')\iota_{c'}(c)=Id_{F(c\ot c')}.$$

[ Let $A$ be an associative algebra and let ${\mathcal{C}}=\text{Bimod}(A)$ denote the tensor category of $A$-bimodules. Then an example of a symmetric $2$-trace is provided by the functor $HH_0(A,-),$ the $0$th Hochschild homology of an $A$-bimodule [@FSS]. We note that for our purposes as outlined below, this example is not very interesting, its only advantage is that it is easy to explain. ]{}

Recall that we denote the subcategory of unital associative algebras in ${\mathcal{C}}$ by $Alg({\mathcal{C}})$. We denote the multiplication morphism of an algebra object $A\in {\mathcal{C}}$ by $m: A\ot A\rightarrow A$ and its unit morphism by $u: 1\rightarrow A$. Given an algebra $A\in Alg({\mathcal{C}})$ and a symmetric 2-trace $F: {\mathcal{C}}\to {\text{Vec}}$, we define a cyclic object in ${\text{Vec}}$ as follows.

Let $$C_n(A)=F(A^{\ot n+1}), n \geq 0.$$ We define the cyclic structure on $C_n (A)$ by

-   $\tau_n = F({\mathfrak{a}}_{1,n}^{n,1})\circ\iota_A(A^{\ot n})$,

-   $\delta_i = F({\mathfrak{a}}^{n-1,1}_{i,1,n-i-1})\circ F(Id^{\ot i}\ot m\ot \Id^{\ot n-i-1})\circ F({\mathfrak{a}}_{n,1}^{i,2,n-i-1})$, for $0 \leq i \leq n-1$,

-   $\delta_{n} 