Let $M$ be some $R$-module and $n \geq 1$ be some positive integer. The cyclic group $\mathbb{Z}/n\mathbb{Z}$, with a chosen generator $t$, acts on $M^{\otimes n}$ via $t(m_1 \otimes \dotsc \otimes m_n) = m_2 \otimes \dotsc \otimes m_n \otimes m_1$. We may mod out the action to obtain some $R$-module $Z_n(M) := M^{\otimes n}/(\mathbb{Z}/n\mathbb{Z})$. Does this $R$-module have a common name? Where can I find some infos about it?
Background. I think that $Z(M):=\bigoplus_{n \geq 1} Z_n(M)$ might be a categorification of the power series $\sum_{n \geq 1} z^n/n = -\log(1-z)$, which is the exponential generating function of the species of cycles.
$Z_n\left(M\right)$ is the $n$-th homogeneous component of the $0$-th cyclic homology of the tensor algebra $T\left(M\right)$.
There is a nice expository note about this by Clas Löfwall, including the relation to the logarithm: Clas Löfwall, Cyklisk homologi, 28 Aug 2012. Despite the title, it is in English.