the definitions of cyclic and cocyclic modules from [@Connes-cyclic] and [@loday].

Recall that the simplicial category $\Delta$ has as its objects non-negative integers considered as totally ordered sets $[n]=\{0,1,\cdots,n\}$ and its morphisms are non-decreasing functions $[n]\to[m]$. A simplicial module is a contravariant functor from $\Delta$ to ${\text{Vec}}$. Similarly, a cosimplicial module is a covariant functor. By keeping the same objects and adding cyclic permutations we obtain Connes cyclic category $C$. A cyclic module is again a contravariant functor from $C$ to ${\text{Vec}}$, while a cocyclic module is a covariant one.

More explicitly, a [*cosimplicial module*]{} is given by the data $(C_{n},\delta_{i}, \sigma_{i})$ where $\{C_{n}\}$, $n\geq 0$ is a sequence of vector spaces over the field $k$. The maps $\delta_{i}:C^{n}\rightarrow C^{n+1}$ are called cofaces, and $\sigma_{i}:C^{n}\rightarrow C^{n-1}$ are called codegeneracies. These are $k$-linear maps satisfying the following cosimplicial relations: $$\begin{aligned}
\label{rel1}
\begin{split}
\delta_{j}  \delta_{i} &= \delta_{i} \delta_{j-1},  \quad   i <j,\\
\sigma_{j}  \sigma_{i} &= \sigma_{i} \sigma_{j+1},    \quad i \leq j,\\
\sigma_{j} \delta_{i} &=
 \begin{cases}
\delta_{i} \sigma_{j-1},   &
 i<j\\
\text{Id},   &   i=j \,\,\text{or}\,\, i=j+1,\\
\delta_{i-1} \sigma_{j},  & i>j+1.
\end{cases}
\end{split}\end{aligned}$$

A [*cocyclic module* ]{} is a cosimplicial module equipped with the extra morphisms $\tau_n :C^n\rightarrow C^n$, called cocyclic maps such that the following 