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Stream: learning: questions

Topic: differential forms in synthetic differential geometry

Max New (Aug 20 2022 at 21:29):

I learned some basic stuff about differential forms a long time ago when I was an undergrad and took a course where we went through Spivak's "Calculus on Manifolds". Every once in a while I try to learn about them in synthetic differential geometry, because I'm told that it is conceptually much simpler and I'm interested in synthetic techniques.

I found a definition in Anders Kock's "Synthetic Geometry on Manifolds" that is very conceptually appealing (Definition 3.1.1 on page 93): a differential k-form on a manifold M with values in a vector space W is just a function from infinitesimal k-simplices in M to W that is 0 if any two vertices of the simplex are equal. Here an infinitesimal k-simplex is a function from "the" infinitesimal k-simplex, i.e., the set of tuples of k points in R that are pair-wise infinitesimally close to each other. This seems like a very natural definition since it is basically literally an infinitesimal notion of volume.

But I don't understand why this definition implies automatically that any differential form is alternating (Theorem 3.1.5) i.e., that if I swap two vertices of a simplex that that reverses the sign of the value of the form. I tried to follow the proof and it came down to something called the "Taylor principle" (page 29) that I didn't quite grasp. When I learned about differential forms analytically, the alternating was motivated by the fact that we were talking about oriented volume and so swapping arguments meant changing the orientation, but I am very curious why the orientedness seems to be an inevitable consequence of the notion of infinitesimal volume? Maybe someone could help me with the infinitesimal geometric intuition here?

John Baez (Aug 21 2022 at 16:08):

Maybe it's related to this: a multilinear map from a vector space to another vector space is alternating iff it vanishes whenever two arguments are equal.

John Baez (Aug 21 2022 at 16:09):

For example say we have $f: V \times V \to W$ such that

$f(v,v) = 0$

for all $v \in V$. Then we have

$0 = f(v+v', v+v') = f(v,v) + f(v,v') + f(v',v) + f(v',v') = f(v,v') + f(v',v)$

so

$f(v,v') = -f(v',v)$

John Baez (Aug 21 2022 at 16:11):

In Kock's approach, the condition $f(v,v) = 0$ is being replaced by the similar condition that your function from infinitesimal k-simplices in M to W "is 0 if any two vertices of the simplex are equal".