a manifold with corners is needed which we recall now based on [@schommerpries Section 3.1.1]: An *$n$-dimensional manifold with corners of codimension 2* is defined to be a second countable Hausdorff space $M$ equipped with a maximal atlas of charts $$\begin{aligned}
M\supseteq U  \stackrel{\varphi}{{\longrightarrow}}  V\subset \mathbb{R}^{n-2}\times (\mathbb{R}_{\ge 0})^2.\end{aligned}$$ For a point $x\in M$ the *index* of $x$ is the number of coordinates of $(\text{pr}_{(\mathbb{R}_{\geq 0})^2}\circ \varphi)(x)$ which are equal to 0 for one chart $\varphi$ (and therefore all charts). We define a *connected face* of $M$ as the closure of a maximal connected subset of points of index 1, a *face* as the disjoint union of connected faces and *manifold with faces* as a manifold with corners having the property that every point of index 2 is part of precisely two different connected faces.

Lastly, we define an *$n$-dimensional $\langle 2 \rangle $-manifold* as an $n$-dimensional manifold $M$ with faces equipped with a decomposition $\partial M = \partial_0 M \cup \partial_1 M$ of its topological boundary into faces having the property that $\partial_0 M \cap \partial_1 M$ is the set of corners of $M$. Here $\partial_0 M$ is called the 0-boundary of $M$ and $\partial_1 M$ the 1-boundary of $M$.

