relations hold: $$\begin{aligned}
\label{rel2}
\begin{split}
\tau_{n}\delta_{i}&=\delta_{i-1} \tau_{n-1}, \quad 1\le i\le  n,\\
\tau_{n} \delta_{0}&=\delta_{n},\\
\tau_n \sigma_{i}&=\sigma_{i-1} \tau_{n+1}, \quad 1\le i\le n,\\
\tau_n^{n+1}&=\Id.\\
\end{split}\end{aligned}$$

In a dual manner, one can define a cyclic module as a simplicial module with extra cyclic maps. More precisely, a cyclic module is given by the data $(C_{n},\delta_i, \sigma_{i}, \tau_{n})$, where $C_{n}$, $n\geq 0$ is a $k$-vector space and $\delta_{i}: C_{n}\rightarrow C_{n-1},  \quad \sigma_{i}:C_n \ra C_{n+1},  \quad 0 \leq i \leq n$, and $\tau_n:C_n \ra C_n$, are called faces, degeneracies and cyclic maps respectively. These are $k$-linear maps, satisfying the following relations: $$\begin{aligned}
\label{rel11}
\begin{split}
\delta_{i} \delta_{j}&=  \delta_{j-1} \delta_{i},  \quad i <j,\\
\sigma_{i}  \sigma_{j}&= \sigma_{j+1} \sigma_{i}, \quad i \leq j,\\
\delta_{i} \sigma_{j} &=
 \begin{cases}
\sigma_{j-1} \delta_{i},   & i<j,\\
\Id,    & i=j \,\,\text{or}\,\, i=j+1,\\
\sigma_{j} \delta_{i-1},  & i>j+1.
\end{cases}
\end{split}\end{aligned}$$ and $$\begin{aligned}
\label{rel22}
\begin{split}
\delta_{i}\tau_n&= \tau_{n-1}\delta_{i-1}, \quad 1\le i\le n,\\
\delta_{0} \tau_n&= \delta_{n}\\
\sigma_{i} \tau_n&= \tau_{n+1}  \sigma _{i-1},\quad 1\le i\le n,\\
\tau_n^{n+1}&= \Id. \\
\end{split}\end{aligned}$$ One notes that the relation $\sigma_{0} \tau_n = \tau_{n+1}^2  \sigma_{n}$ that is usually listed along with the above is an extra relation [@loday section 5.2] which can be obtained from $\tau^{n+1}_n=\Id$ and $\sigma_{i} \tau_n= \tau_{n+1}  \sigma _{i-1}$. Similarly for a cocyclic module $\tau_n 