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Stream: event: Topos Colloquium

Topic: David Spivak: “Poly: a category of remarkable abundance”

Tim Hosgood (Feb 04 2021 at 12:07):

Hi all, today is the day of the inaugural Topos Institute Colloquium talk, and we have David himself speaking. It starts at 17:00 UTC and is on Zoom, but will be live streamed to the Topos Institute YouTube account too. Links will be posted here closer to the time, but you can find more information on https://topos.site/seminars .

Tim Hosgood (Feb 04 2021 at 12:08):

here’s the abstract:

The category Poly, of polynomial functors in one variable, is striking. From a purely theoretical standpoint, its comonoids are exactly categories and its monoids generalize operads. Consider: categories and generalized operads falling out of a single setting as the comonoids and monoids is already striking.

But Poly is also impressive from an applications-oriented standpoint. Indeed, it naturally contains database schemas, database instances, and data migration functors; cellular automata like Conway's Game of Life; many kinds of dynamical systems — discrete-time, real-time, etc. — and very general sorts of interactions between them.

In this talk I'll explain all these ideas as well as some of the many avenues for future work. I'll start with the basics but move quickly, with the talk acting more as a calling card — saying "reach out if this interests you" — rather than as a full-fledged explanation of any particular aspect. My goal is simply to show off the remarkable elegance and applicability of this abundant category.

Tim Hosgood (Feb 04 2021 at 16:40):

the YouTube live stream is https://youtu.be/Cp5_o2lDqj0 and will go live when the talk starts, in 20 minutes

Tim Hosgood (Feb 04 2021 at 16:54):

live in 5!

Tarmo Uustalu (Feb 04 2021 at 17:05):

Where is the Zoom link?

Tim Hosgood (Feb 04 2021 at 17:08):

sorry, I forgot to post that here: https://topos-institute.zoom.us/j/97151871676?pwd=eDBKSEV2M0x6Qm1UWnVSZURVWWx3QT09

Matthias Hutzler (Feb 04 2021 at 17:23):

Are the slides already available?

Tim Hosgood (Feb 04 2021 at 17:24):

not yet I'm afraid, but I'll ask David for a link as soon as the talk is done

Eric Bond (Feb 04 2021 at 17:59):

Where was that link for the Poly workshop?

Nathanael Arkor (Feb 04 2021 at 18:00):

Here.

Bruno Gavranović (Feb 04 2021 at 18:03):

The morphisms of polynomials are what I understand to be something like dependent lenses. But there's an interesting generalization of lenses called optics - do you know how optics fit into the story of Poly?

Peter Arndt (Feb 04 2021 at 18:08):

I will just put my end of talk clapping here: :clap: I loved it!

Joachim Kock (Feb 04 2021 at 18:10):

Beautiful talk, David. And full of hope!

fosco (Feb 04 2021 at 18:14):

Thanks @David Spivak for the wonderful talk. Let me say I was driven to Mathematics by similar reasons, and also that after your talk I have one more argument to question monogamy :grinning:

Now for the question: a few days ago I posted a question here on zulip that evidently intersects the theory of $\sf Poly$: here.

As you can see, I've been trying to define a "2-rig with a differential" for quite some time now, and I have plenty of questions.
If I understand well, $\sf Poly$ is a rig category, and I wonder what could interesting derivations on it be, and what could they be useful for. I have something in mind!

Joachim Kock (Feb 04 2021 at 18:51):

There is a derivative for polynomial functors, but it is only functorial in cartesian natural transformations (and thereby it goes a bit in other directions than the category Poly of David's talk). This derivative corresponds to ordinary derivation: $D y^n = n y^{n-1}$, extended linearly. So what is $n-1$? (remembering that $n$ is a set not a number!) Well, you need to remove one element, and you don't know which one to remove, but luckily you have $n$ choices and you can take them all, because there is a coefficient $n$ in front. So to write it more invariantly, the formula is

$D y^n = \sum_{i\in n} y^{n\setminus \{i\}}$

Now extend linearly and consider the representing maps: if the polynomial functor $P$ is represented by $E\to B$ (here $B$ is the set of positions and $E$ the set of directions, in David's terminology), then the derivative $DP$ is represented by $E\times_B E \setminus \Delta \to E$ (fibre product minus the diagonal).

This derivative is the same as for species and analytic functors. In fact (finitary) polynomial functors are a special case of species, namely the socalled flat species. They are those for which the symmetric-group actions are free. (If you upgrade from sets to groupoids, then finitary polynomial functors are equivalent to species (stuff types).) (And monoids for the composition monoidal structure are (sigma-cofibrant) operads — in the cartesian part, not in the big category Poly David talks about).

fosco (Feb 04 2021 at 20:15):

I have to think about this. Thanks in advance Joachim!

fosco (Feb 04 2021 at 20:36):

In that paper about differential 2-rigs I have defined something that resembles a category of polynomials; I need some time to reorder the notes, but I'd like to see how does it connect with $\sf Poly$

Matteo Capucci (he/him) (Feb 05 2021 at 08:08):

That definition of derivative is so beautiful! It really gives meaning to that $n$ coefficient.
Directions get promoted to positions, like 'changes' of a function $f$ are promoted to 'values' of its derivative $f'$, and the fact that the $i$-th direction (now a position) has $n\setminus\{i\}$ as set of directions is remindful of $dx_i^2 = 0$.

David Michael Roberts (Feb 05 2021 at 10:34):

That's pretty amazing @Joachim Kock , I just had to share it (and said I learned it from you) on Twitter.

Joachim Kock (Feb 05 2021 at 13:19):

I learned it from André Joyal. I find the polynomial-functor interpretation even nicer than the one in species, also for the reasons Matteo expressed. And it fits very well with the dynamical viewpoint promoted by David. Differentiation of species is also due to Joyal, or course – it's in the original species paper Une théorie combinatoire des séries formelles (1981).

David Spivak (Feb 09 2021 at 20:13):

Matthias Hutzler said:

Are the slides already available?

They're here now: http://www.dspivak.net/talks/Topos20210204.pdf

Olli (Feb 26 2021 at 08:10):

I have seen mentions of a book about Poly, available on Github, but I could not find anything searching for it. Is this book publicly available, and if so could someone share a link?

Tim Hosgood (Feb 26 2021 at 08:59):

https://github.com/ToposInstitute/poly/blob/main/Book-Poly.pdf

Olli (Feb 26 2021 at 09:06):

Thanks!