$M$, and the union of the faces is the *boundary* $\partial M $ of $M$.

Now suppose each $p\in M$ belongs to precisely $l(p)$ faces. In this case, a *smooth $m$–dimensional $\langle n\rangle$–manifold* is a manifold with corners $M$, together with an ordered $n$–tuple of faces $(\partial_1M,\dots,\partial_nM)$ which satisfy the following conditions:

-   $\partial_1M\cup\dots\cup\partial_nM=\partial M$,

-   $\partial_iM\cap\partial_jM$ is a face of $\partial_iM$ and $\partial_jM$ for all $i\neq j$.

In which case, for any non-empty subset $J\subset \{1,\dots,n\}$, we will write$$\partial_JM=\bigcap_{j\in J}\partial_jM.$$

Now we introduce some more notation. Let $n$ be a non-negative integer and let ${\textbf{d}}=(d_0,\dots,d_n)$ be an $(n+1)$–tuple of non-negative integers. Write ${\mathbb{R}}_+=[0,\infty)$ and $$\begin{array}{rclll}{\mathbb{E}}^{\textbf{d}}&=&{\mathbb{R}}^{d_0}\times{\mathbb{R}}_+\times {\mathbb{R}}^{d_1}\times{\mathbb{R}}_+\times\dots\times{\mathbb{R}}_+\times{\mathbb{R}}^{d_n},&&\\ {\mathbb{E}}^{\textbf{d}}[a:b]&=&{\mathbb{E}}^{(d_a,\dots,d_{b-1})},&&\text{for $0\leq a<b\leq n+1$.}\end{array}$$We introduce a standard subdivision of the boundary of ${\mathbb{E}}^{\textbf{d}}$ into the following faces: $$\partial_i{\mathbb{E}}^{\textbf{d}}={\mathbb{R}}^{d_0}\times{\mathbb{R}}_+\times\dots{\mathbb{R}}^{d_{i-1}}\times\{0\}\times{\mathbb{R}}^{d_i}\times\dots\times {\mathbb{R}}_+\times{\mathbb{R}}^{d_n},$$for $i=1,\dots,n$, so that ${\mathbb{E}}^{\textbf{d}}$ is a $(d_0+\dots+d_n+n)$–dimensional $\langle n\rangle$–manifold.

\[def:neat\]A *neat immersion (embedding)* of a smooth $m$–dimensional $\langle n\rangle$–manifold $M$ is a smooth immersion (embedding) $\iota:M \looparrowright {\mathbb{E}}^{\textbf{d}}$ such that for all $l$, corner points of codimension $l$ in $M$ are sent to corner points of codimension $l$ in ${\mathbb{E}}^{\textbf{d}}$, and for any $J\subset I\subset \{1,\dots n\}$ the intersection $\partial_IM\cap\partial_J{\mathbb{E}}^{\textbf{d}}=\partial_J M$ is perpendicular.

Any compact $m$–dimensional $\langle n\rangle$–manifold admits a neat embedding with respect 