to some ${\mathbb{E}}^{\textbf{d}}$ (see Laures [@MR1781277 2.1.7]). Moreover, a neat immersion can always be improved to a neat embedding by perturbation, possibly after increasing the integers in the tuple ${\textbf{d}}$.

A *flow category *${\mathscr{C}}$ is a category with finitely many objects, and equipped with a function ${|{\cdot}|}\colon {\text{Ob}}({\mathscr{C}}) \to {\mathbb{Z}}$, called the *grading*, satisfying the following:****

1.  ${\text{Hom}}(x,x)=\{{\text{id}}\}$ for all $x\in{\text{Ob}}({\mathscr{C}})$, and for $x\neq y\in{\text{Ob}}({\mathscr{C}})$, ${\text{Hom}}(x,y)$ is a smooth, compact $(|x|-|y|-1)$–dimensional $\langle |x|-|y|-1\rangle$–manifold which we denote by ${\mathcal{M}}(x,y)$.

2.  For $x,y,z\in {\text{Ob}}({\mathscr{C}})$ with ${|{z}|}-{|{y}|}=m$, the composition map restricts to $$\circ\colon \mathcal{M}(z,y) \times \mathcal{M}(x,z) \to \mathcal{M}(x,y),$$ which is an embedding into $\partial_m\mathcal{M}(x,y)$. Furthermore, $$\circ^{-1}(\partial_i \mathcal{M}(x,y)) = \left\{ \begin{array}{lr}
                                                 \partial_i \mathcal{M}(z,y)\times \mathcal{M}(x,z) & \mbox{for }i<m \\
                                                 \mathcal{M}(z,y)\times \partial_{i-m}\mathcal{M}(x,z) & \mbox{for }i>m
                                                \end{array}
    \right.$$

3.  For $x\not= y\in {\text{Ob}}({\mathscr{C}})$, $\circ$ induces a diffeomorphism $$\partial_i\mathcal{M}(x,y) \cong \coprod_{z,\,{|{z}|}={|{y}|}+i} \mathcal{M}(z,y) \times \mathcal{M}(x,z).$$

The manifold $\mathcal{M}(x,y)$ is called the *moduli space from $x$ to $y$*, and we also set $\mathcal{M}(x,x)=\emptyset$.**

Let ${\mathscr{C}}$ be a flow category and write $$k=\min\{|x|\,:\,x\in{\text{Ob}}({\mathscr{C}})\},\quad n=\max\{|x|\,:\,x\in{\text{Ob}}({\mathscr{C}})\}-k.$$ Suppose ${\textbf{d}}=(d_k,\dots,d_{n+k})$ is an $(n+1)$–tuple of non-negative integers. A *neat immersion (resp. neat embedding)* $\iota$ of a flow category ${\mathscr{C}}$ relative to ${\textbf{d}}$ is a collection of neat immersions (resp. neat embeddings) $\iota_{x,y}:{\mathcal{M}}(x,y)\looparrowright{\mathbb{E}}^{\textbf{d}}[|y|:|x|]$ such that for all 