if $\sigma$ is in the domain of $X_i$ for some $i$, **collapsible** if it is in the image of $X_i$ for some $i$, and **critical** otherwise.\

To get a better intuition for the above definitions, one might also consider the following (equivalent) formulation. Let $f:{\mathcal{K}}\rightarrow {\mathbb{R}}$ be a map such that for all $\sigma \in {\mathcal{K}}_i$,

1.  $|\{\tau \in {\mathcal{K}}_{i+1} \mid \sigma \subseteq \overline{\tau} \text{ and } f(\sigma) \geq f(\tau)\}| \leq 1$; \[dm1\]

2.  $|\{\tau \in {\mathcal{K}}_{i-1} \mid \tau \subseteq \overline{\sigma} \text{ and } f(\sigma) \leq f(\tau)\}| \leq 1$. \[dm2\]

One calls $f$ a discrete Morse function, and it can be thought of as being analogous to a Morse function in the classical sense. Cells for which \[dm1\] is an equality are redundant, for which \[dm2\] is an equality are collapsible, and for which both are not equalities are critical. It is a fact [@Fo1] that these are actually the only three cases. One immediately notes that according to this formulation, the classification of cells does not fully depend on the explicit values of the function $f$. In other words, postcomposition with any monotone function on ${\mathbb{R}}$ does not change the resulting classification of cells. One then constructs a discrete 