general results about simplification of framed flow categories. Essentially, these results can be taken as saying that each framed flow category is move-equivalent to a framed flow category realizing the simplest possible cochain complex giving the correct cohomology.

Framed flow categories {#sec:ffcs}
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The definition of a flow category was originally given by Cohen, Jones and Segal [@MR1338869] but the version we are working with is based on the definition by Lipshitz and Sarkar [@MR3230817]; the equivalence of the two approaches is shown in [@MR3230817 Prop. 3.27].

Before we can give the definition of a flow category, we will need a sharpening of the concept of a smooth manifold with corners. This refinement was originally defined by J[ä]{}nich [@MR0226674] and was further devloped by Laures [@MR1781277]. We refer the reader to these sources for a fuller treatment.

Let $M$ be a smooth $m$–dimensional manifold with corners. For each $p\in M$ there is an integer $l(p)$ such that an open neighbourhood of $p$ is homeomorphic to $({\mathbb{R}}_+)^l\times {\mathbb{R}}^{m-l}$, with $p$ corresponding to $0\in ({\mathbb{R}}_+)^l\times {\mathbb{R}}^{m-l}$. We say $p$ is a *corner point of codimension $l$*. The closure of any connected component of the set $\{p\in M\,|\,l(p)=1\}$ is called a *face* of 