I am currently studying augmented simplicial sets with some additional degeneracies. I was wondering if it is a structure that was already identified somewhere. This additional data is a family of applications $s_i : S_n \to S_{n+1}$ for $n \geq 0$ and $0 \leq i \leq n$ satisfying the following axioms:

$$ \partial_i s_j = \begin{cases} s_{j-1} \partial_{i-1} & j < i \\ Id & j = i \\ s_j \partial_i & j > i \end{cases} \qquad \sigma_i s_j = \begin{cases} s_j \sigma_{i-1} & j < i \\ s_j s_j & j = i \\ s_{j+1} \sigma_i & j > i \end{cases} \qquad s_is_j = s_{j+1} s_{i} \quad i \leq j $$

Some intuition about this structure: it is well known that in the opposite of the augmented simplex category $\Delta_a^{op}$, the object $\mathbf{1}$ is in some sense the universal monoid. The structure I am presenting here are presheaves over a category $\Delta_+$, where $\mathbf{1}$ would be in $\Delta_+^{op}$ the universal "monoid with a morphism towards the unit of the tensor product".

For those interested, this is also heavily related to cubical sets: they correspond to cubical sets where you throw away the connections $\Gamma^+$ and the faces $\partial^+$.