objects $x,y,z$ and points $(p,q)\in {\mathcal{M}}(x,z)\times{\mathcal{M}}(z,y)$,$$\iota_{x,y}(q\circ p)=(\iota_{z,y}(q),0,\iota_{x,z}(p)).$$

If $\iota$ is a neat immersion of a flow category ${\mathscr{C}}$ relative to ${\textbf{d}}$, then a *framing* ${\varphi}$ of $({\mathscr{C}},\iota)$ is a collection of immersions$${\varphi}_{x,y}:{\mathcal{M}}(x,y)\times[-{\varepsilon},{\varepsilon}]^A\looparrowright {\mathbb{E}}^{\textbf{d}}[|y|:|x|],\qquad A=d_{|y|}+\dots+d_{|x|-1},$$extending $\iota_{x,y}$, such that for all objects $x,y,z$ and points $(p,q)\in {\mathcal{M}}(x,z)\times{\mathcal{M}}(z,y)$,$${\varphi}_{x,y}(q\circ p,t_1,t_2,\dots, t_A)=({\varphi}_{z,y}(q,t_1,t_2,\dots,t_B),0,{\varphi}_{x,z}(p,t_{B+1},\dots,t_A)),$$where $B= d_{|y|}+\dots+d_{|z|-1}$.

Note that a (framed) neat immersion of a flow category relative to some ${\textbf{d}}$ can be improved to a (framed) neat embedding relative to some ${\textbf{d}}'\geq{\textbf{d}}$ by perturbation.

A *framed flow category* is a triple $({\mathscr{C}},\iota,{\varphi})$, where $({\mathscr{C}},\iota)$ is a flow category with neat immersion, relative some ${\textbf{d}}$, and ${\varphi}$ is a framing for $({\mathscr{C}},\iota)$.

One can associate to a framed flow category $({\mathscr{C}}, \iota,{\varphi})$ a geometric realisation as a finite cell complex $|{\mathscr{C}}|_{\iota,{\varphi}}=|{\mathscr{C}}|$ (see [@MR3230817 Definition 3.23] for precise details of the version we are using). The construction of $|{\mathscr{C}}|$ requires several auxilliary choices, one such being an integer $C$ such that $C>|x|$ for all $x\in{\text{Ob}}({\mathscr{C}})$. To construct $|{\mathscr{C}}|$, one takes for each object $x$ of ${\mathscr{C}}$ a $(C-m)$–dimensional cell $\mathcal{C}(x)$, where $|x|=m$. These cells are glued together using the data of the framed moduli spaces (essentially via the Thom construction).

The homotopy type of $|{\mathscr{C}}|$ depends on the framed 