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Stream: theory: applied category theory

Topic: chainlets

Spencer Breiner (May 28 2021 at 22:23):

All the talk about polytopes the last few days has me reminiscing about a fascinating topic that I ran into at the end of my time as an undergrad at Berkeley. It is a modern approach to the development of vector calculus, and I've always thought the approach deserves to be better known. I'd love it if some of these ideas could be worked into a graphical calculus.

Spencer Breiner (May 28 2021 at 22:25):

I see the topic has already been discussed at the Café, but I think it was before I started lurking.

Spencer Breiner (May 28 2021 at 22:28):

The context of the work is topological vector spaces, and the main idea is to directly construct a rich vector space of domains which has differentials/differential forms as it's dual. This is in contrast to the usual approach, where we start with some relatively simple class of domains (e.g., intervals), describe integration over those, and obtain a richer (too rich?) collection of domains as the double dual.

Spencer Breiner (May 28 2021 at 22:49):

The fundamental desideratum is that the boundary map should be continuous, as this will allow us to define the differential via duality, making Stokes' theorem definitional. This is not entirely trivial.

First imagine a domain $D_t=[-t,t]$, which obviously has boundary $\{\pm t\}$. As $t\to 0$, we expect $||D_t||\to 0$ as well; in particular, for any continuous function $\int_{D_t} f\ dx\to 0$, so this is a kind of positive definiteness condition.

Spencer Breiner (May 28 2021 at 23:10):

What about $\partial D_t$?

Spencer Breiner (May 28 2021 at 23:28):

A priori, we have a few reasonable interpretations of what to do when the endpoints merge in the limit. We could reasonably expect $\partial D_{[0,0]}=D_{\{0\}}$, since $\partial [0,0]=\partial \{0\}$ as subspaces of $\mathbb{R}$. That would cause problems, though, since $||D_{\{0\}}||=||D_{\{t\}}||=\frac{1}{2}(||D_{\{-t\}}||+||D_{\{t\}}||)$. Setting $||\partial D_t||=0$ is allowed, but boring.

Spencer Breiner (May 28 2021 at 23:33):

We get around this by introducing orientation. If we set $\partial D_t=D_{\{t\}}-D_{\{-t\}}$ there's no problem: the opposites annihilate when (and only when) the interval has zero length.

Spencer Breiner (May 28 2021 at 23:48):

We might expect that for disjoint domains $D$ and $D'$, we should have $\int_{D\cup D'}f\ dx=\int_D f\ dx+\int_{D'} f\ dx$, but this turns out to be too strong. If $||\partial D_t||=||D_{\{t\}}||+||-D_{\{-t\}}||$, we would have a discontinuity in the norm as $t\to 0$.

Spencer Breiner (May 28 2021 at 23:52):

We introduce context through an interaction between the norm and translations:
If $D$ can be decomposed as a vector-separated difference $D=T_v D_0 - T_{-v} D_0$, then $||D||\leq ||v||\cdot||D_0||$. This ensures continuity in the limit as $v\to 0$, so the boundary of a small interval tends to zero.

Spencer Breiner (May 29 2021 at 00:06):

One interesting feature of this approach is that we can weight regions in order to define Dirac deltas directly, rather than recovering them in the double dual. In particular, the limit $L_*$ of the weighted intervals $L_t=t^{-1}\cdot D_t$ is non-zero, and represents a Dirac delta at 0. Moreover, the boundary of $L_*$ is also non-zero, and represents an infinitesimal dipole: $||\partial L_*|| =\lim_t t\cdot||D_{\{t\}}-D_{\{-t\}}||$.

Spencer Breiner (May 29 2021 at 00:09):

Notice that the factor of $t^{-1}$ prevents the cancellation of opposite orientations.

Amar Hadzihasanovic (May 29 2021 at 09:05):

Thanks, I was not aware of this work, it seems super interesting!

Jacques Carette (May 29 2021 at 13:57):

Is there a readable introduction? The Café link has messages from J. Harrison herself saying that the linked writeup is obsolete. But going to her publications page or her google scholar page does not enlighten me either.

Amar Hadzihasanovic (May 29 2021 at 15:12):

This paper seems to be the most up-to-date introduction!

Jacques Carette (May 29 2021 at 16:38):

Well, it is a paper, I agree. An 'introduction'? Hmm, I'm rather less sure about that...

Amar Hadzihasanovic (May 29 2021 at 17:30):

Well, it's an introduction in the sense that it develops the theory from scratch giving all the definitions. Considering it's quite recent research, I doubt there's anything more “friendly”.

Spencer Breiner (Jun 01 2021 at 16:04):

Glad someone liked it. As she discusses in the paper Amar linked, the presentation starting from discrete domains is much easier, but I actually find some of the earlier presentations in terms of polyhedral chains more charming; it's a nice feature of the theory is that you can build it starting from standard geometric objects.

Spencer Breiner (Jun 01 2021 at 16:10):

I brought it up here because the intellectual ferment in a parallel thread included diagrammatic definitions for various polyhedral constructions, including the boundary. There everything is written down in terms of posets, but it seems like one could enrich in a related category (posets over $\mathbb{R}^+$?) to add norms to into the mix. I don't have time to dig into it, but I would love it if someone found a connection.