## Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9

Dear Kenneth,

Mulling over recent discussions put me in a pensive frame of mind and my thoughts led me back to my first encounter with category theory.  I came across the term while reading and I didn’t fully understand it.  But I distinctly remember a short time later catching up with my math TA — it was on the path by the tennis courts behind Spartan Stadium — and asking him about it.

The instruction I received that day was roughly along the following lines.

“Actually . . . we’re already doing a little category theory, without quite calling it that.  Think about the different types of spaces we’ve been discussing in class, the real line $\mathbb{R},$ the various dimensions of real-value spaces, $\mathbb{R}^n, \mathbb{R}^m,$ and so on, along with the various types of mappings between those spaces.  There are mappings from the real line $\mathbb{R}$ into an $n$-dimensional space $\mathbb{R}^n$ — we think of those as curves, paths, or trajectories.  There are mappings from the plane $\mathbb{R}^2$ to values in $\mathbb{R}$ — we picture those as potential surfaces over the plane.  More generally, there are mappings from an $n$-dimensional space $\mathbb{R}^n$ to values in $\mathbb{R}$ — we think of those as scalar fields over $\mathbb{R}^n$ — say, the temperature at each point of an $n$-dimensional volume.  There are mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$ and mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$ where $n$ and $m$ are different, all of which we call transformations or vector fields, depending on the use we have in mind.”

All that was pretty familiar to me, though I had to admire the panoramic sweep of his survey, so my mind’s eye naturally supplied all the arrows for the maps he rolled out.  A curve $\gamma$ through an $n$-dimensional space would be typed as a function $\gamma : \mathbb{R} \to \mathbb{R}^n,$ where the functional domain $\mathbb{R}$ would ordinarily be regarded as a time dimension.  A mapping $\alpha$ from the plane to a real value would be typed as a function $\alpha : \mathbb{R}^2 \to \mathbb{R},$ where we might be thinking of $\alpha(x, y)$ as the altitude of a topographic map above each point $(x, y)$ of the plane.  A scalar field $\beta$ defined on an $n$-dimensional space would be typed as a function $\beta : \mathbb{R}^n \to \mathbb{R},$ where $\beta(x_1, \ldots, x_n)$ is something like the pressure, the temperature, or the value of some other dependent variable at each point $(x_1, \ldots, x_n)$ of the $n$-dimensional volume.  And rounding out the story, if only the basement and ground floor of a towering abstraction still under construction, we come to the general case of a mapping $f$ from an $n$-dimensional space to an $m$-dimensional space, typed as a function $f : \mathbb{R}^n \to \mathbb{R}^m.$

To be continued …

### 2 Responses to Differential Logic, Dynamic Systems, Tangent Functors • Discussion 9

1. Kenneth Lloyd says:

There is a specific type of ‘arrow’ (morphism) that shows the transformation of a category (usually a capital letter) to a metric space (in ‘blackboard font’) or one metric space to another.  In LaTeX, it is the \mapsto command.  The concepts behind mapping and projection are subtle and powerful.  It is taking something abstract and projecting it onto the meaningful metric space — a kind of context.

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