Polynomials as $n$-cells

I want to articulate a way in which it almost¹ seems possible to interpret almost² any multivariate polynomial function $f:\R^n \to \R$ as a $n$-dimensional cell in an $\omega$-category. In a single sentence, the crux of the interpretation is:

Read the graph of $f$ as a string diagram.

In what follows we'll assume all functions mentioned are polynomials unless specified otherwise.

Example

Let $f : \R^2 \to \R$ be \[f(x,y) = x^3 + y - 3x\] We can graph it and interpret it as a string diagram like so:

The region where the function $f$ is negative is considered an object $C$ of the $\omega$-category. The region where $f$ is positive is called the object $D$. The points where $f$ is zero, but $f_x$, the $x$-derivative of $f$, is positive, form two curves labelled $F$ in the diagram: we interpret these as a morphism $F : C \to D$. Places where $f = 0$ and $f_x < 0$ we interpret as a morphism $U : D \to C$.

Finally, the points where $F$ and $U$ come together in the diagram are viewed as 2-cells. At the point labelled as the 2-cell $\eta : \rid_{C} \to UF$, the function $f$ has the property that $f = 0$ and $f_x = 0$, but $f_{xx} > 0$ and $f_y > 0$. At the point labelled $\epsilon :FU \to \rid_{D}$, we have $f = 0$ and $f_x = 0$, but $f_{xx} < 0$ and $f_y > 0$.

The intended interpretation of $f = x^3 + y - 3x$ is the composite 2-cell \[ \epsilon_F \o F \eta : F \to F\] familiar from the triangle identities of an adjunction.

Criticality

The crucial thing is to define which points of the graph of $f$ count as 'exceptional', 'interesting', 'critical', i.e. where higher dimensional cells are happening. There are various levels of criticality, one for each dimension. For a point to be $k$-critical means that belongs to a $k$-cell in the graph of $f$.

We first have to define an auxiliary function $D^k(f) : \R^n \to \R$, for any $f(x_1, \ldots, x_n) : \R^n \to \R$ and integer $k$ such that $1 \le k \le n$. We say:

$D^1(f) = f$
$D^{k+1}(f) = \displaystyle\det_{1\le ij \le k} {\partial D^i(f) \over \partial x_j}= \left| \begin{array}{ccc} {\partial \over\partial x_1} D^1(f)&\ldots & {\partial \over\partial x_k}D^1(f) \\ \vdots & \ddots & \vdots \\ {\partial \over\partial x_1} D^k(f)&\ldots & {\partial \over\partial x_k}D^k(f) \\ \end{array}\right|$

We say a point $p\in \R^n$ is $k$-critical in $f$ if $D^{\ell}(f) = 0$ for all $\ell \in 1,\ldots, k$.

We say a point is exactly $k$-critical if it is $k$-critical and not $(k+1)$-critical.

Checking Back With the Example

We can see that this definition lines up with what we were saying in the example above. The morphisms are points that are exactly 1-critical — the points where $D^1(f) = f = 0$, but where \[D^2(f) = \left| {\partial D^1(f) \over \partial x_1 }\right| = f_x \ne 0 \]

Similarly, the 2-cells are the points that are exactly 2-critical — where $f_x = 0$, but where \[ D^3(f) = \left| \begin{array}{cc} {\partial \over\partial x_1} D^1(f) & {\partial \over\partial x_2}D^1(f) \\ {\partial \over\partial x_1} D^2(f) & {\partial \over\partial x_2}D^2(f) \\ \end{array} \right| \] \[= \left| \begin{array}{cc} {\partial \over\partial x} f & {\partial \over\partial y}f \\ {\partial \over\partial x} f_x & {\partial \over\partial y}f_x \\ \end{array} \right| = \left| \begin{array}{cc} f_x & f_y \\ f_{xx} & f_{xy} \\ \end{array} \right| = \left| \begin{array}{cc} 0 & f_y \\ f_{xx} & f_{xy} \\ \end{array} \right| = -f_{xx}f_y \ne 0 \]

Cells of the Putative $\omega$-Category

An $n$-cell is a function $f:\R^n \to \R$ that has no $(n+1)$-critical points.

An $n$-cell is trivial if doesn't have any $n$-critical points.

We expect that trivial cells act like equivalences: the domain and codomain of a trivial cell can be identified. However, we haven't even defined what the domain and codomain of cells are!

We'd like to say something like: the domain of a morphism $f : \R \to \R$ is $f(-\infty)$ and the codomain of $f$ is $f(\infty)$. This isn't precise, but since we said that functions that don't change sign are trivial, we identify all positive 0-cells together, (calling them collectively $C$ in the example above) and all negative 0-cells together (calling them $D$ in the example above). Therefore it is more or less sensible to ask for $f(\infty)$ as being either $C$ or $D$ according to the sign $f$ asympotically reaches if you make its argument arbitrarily large.

Questions

But obviously this is no good as a definition of a category if we can't compose cells.