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Since someone was asking about this on Twitter, I tried to make some sense of Section II there. Is there more going on?
Interesting, I don’t remember appreciating that Section II before: it is re-casting the operations that go into the statement of the Hadamard lemma.
Compare to Hadamard’s difference equation here:
The $x$ there (think of it as $x - 0$) is Lawvere’s $\Delta x$
the $f(x) - f(0)$ there is Lawvere’s $\Delta y$,
and $g$ is $g$.
I’ll add cross-link.
I was trying to make up a story for the enquirer about the adjoint cylinder aspect. So the composites $pr_i^{\ast} \circ \diag^{\ast}$ are both idempotent, but does it help to see a two variable function, $g(x,y)$, as sandwiched between $g'(x,y) = g(x,x)$ and $g''(x,y)= g(y,y)$? Some kind of extremes in class of two variable functions with same diagonal?
Why am I reminded of that thing about intrinsic curvature? What is it? At any point, two perpendicular directions of max and min curvature.
Oh, is it infinitesimal cohesion, quality type terrain?
It seems to me that the relation to cohesion in this article is superficial.
After all, that “cylinder” (here) is just the beginning of a cosimplicial ring, not an adjoint triple between categories. Even if one were promote it to an adjoint triple, say by extension/restriction of scalars between categories of modules, then the difference operation (“$\Delta$ in the notation of that section II) still is – while crucial for the argument – not accounted for by the diagrammatics, nor is its appearance in the formulas.
What I see in the article is an example of taking an elementary fact, here the Hadamard lemma, and checking which bare-bones category theoretic structure is needed to formulate it, hence possibly to internalize it elsewhere. And here it’s not much: We need functions depending on either one or two variabables, and the ways to turn these into each other.
To the extent that all this is meant to be about teaching kids (as the article seems to be suggesting), one could argue that it motivates the notion of Cartesian product and of their images under contravariant functors hiding in the formulation of a simle but profound classical lemma.
Now Hadamard’s lemma is a deep statement about smooth differential geometry, I have called it the third of the three magical properties of the smooth category (here) – as such it underlies much of the construction and discussion of the canonical models of synthetic/cohesive differential geometry. But that deep property is not what the discussion in “Unity and Identity…” illuminates. It just appeals to it.
Ok, thanks. Perhaps no deep insight by Marx then:
Near the end of his life, Karl Marx wrote about the foundations of differential calculus. The essence of his line of thought, later rigorously established by Hadamard, yields an effective and simple basis for learning and developing the subject if made explicit.
As I said, Hadamard’s lemma in itself is one of these simple lemmas that are crucially important (like the Yoneda lemma in category theory). If Marx really saw this that would be impressive. It would be fun to see this referenced. But maybe Lawvere got carried away here with attributing insight to Marx?
If someone ever wishes to follow this up, Marx’s Mathematical Manuscripts.
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