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Stream: theory: applied category theory

Topic: Manin--Marcolli

Rich Hilliard (Jun 30 2020 at 15:06):

This paper, seen on twitter, seems to touch on a number of ACT topics (but quickly goes beyond my ability to read):
Homotopy Theoretic and Categorical Models of Neural Information Networks
Yuri I. Manin & Matilde Marcolli
arXiv:2006.15136

Evan Patterson (Jun 30 2020 at 21:18):

Another paper by Marcolli (and also Port) with an ACT flavor is: Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory (arxiv:1502.07796). I haven't read either paper, so I can't say more.

Fabrizio Genovese (Jul 01 2020 at 14:25):

I skimmed through this paper. Clearly the details are a no-no for me, but this is a treasure trove

Fabrizio Genovese (Jul 01 2020 at 14:26):

It goes in a direction which is completely orthogonal to the one I was thinking about to describe these kind of things, which is great, because now I have a lot of new tools to learn and to practice with

Evan Patterson (Jul 02 2020 at 17:02):

Thanks, that looks interesting!

Fatima Afzali (Jul 09 2020 at 08:08):

Absolutely. I've been going through it on my own, since it ties together several of my research interests, but it gets very difficult very fast.

Fabrizio Genovese (Jul 09 2020 at 11:45):

ME

Fabrizio Genovese (Jul 09 2020 at 11:46):

I really need to understand all this gamma spaces stuff

Aleksandar Makelov (Jul 09 2020 at 12:16):

for new people like me, can someone summarize the high-level idea of the paper and how it may be relevant more broadly to act?

Fabrizio Genovese (Jul 09 2020 at 12:17):

"We use badass geometry to describe networks of things, in particular neural networks".

Fabrizio Genovese (Jul 09 2020 at 12:18):

My feeling is that they are really not in contact with the ACT community, so a lot of things are done weirdly and are weirdly motivated (they are not exactly in touch with the practical). Still, their maths is increadible, so the paper can be a treasure trove to pinpoint and apply techniques that can be used in other ACT fields.

Toby Smithe (Jul 09 2020 at 12:18):

Fabrizio Genovese said:

I really need to understand all this gamma spaces stuff

Same. There seem to be a lot of gems in this paper. Manin's work on space in the brain agrees (as far as I can tell) with much of my own thinking, though of course his work is much more rigorously developed than my ideas (yet!).

Oh yeah, I still owe you an e-mail about sheaves and brains, I think, @Fabrizio Genovese ...

Toby Smithe (Jul 09 2020 at 12:19):

Fabrizio Genovese said:

My feeling is that they are really not in contact with the ACT community, so a lot of things are done weirdly and are weirdly motivated (they are not exactly in touch with the practical). Still, their maths is increadible, so the paper can be a treasure trove to pinpoint and apply techniques that can be used in other ACT fields.

Yes, this is my feeling too. It will take me some time, but AFAICT it will be worth studying this work.

Fabrizio Genovese (Jul 09 2020 at 12:19):

We should work together, I'm basically investigating sheaves of lenses and other stuff along with @Matteo Capucci and other people.

Fabrizio Genovese (Jul 09 2020 at 12:19):

My focus is consensus mechanisms, but really the whole point of this thing is developing general techniques that can be used everywhere

Fabrizio Genovese (Jul 09 2020 at 12:19):

I'm happy to have a call as soon as this ACT madness finishes :grinning:

Toby Smithe (Jul 09 2020 at 12:20):

Oh nice, that sounds very interesting, and yes I've started to study consensus myself, both because I am interested in distributed systems (like blockchain!) generally, and also because I am sure that there is consensus-finding in how the brain works. And of course the local/global thing is at the heart of such consensus.

Toby Smithe (Jul 09 2020 at 12:21):

By sheaves of lenses do you mean something like the sheaves in $[\mathcal{C}^{op}, \mathbf{Tamb}]$ ..? Like, lens-valued sheaves..? What is the type of this thing?

Toby Smithe (Jul 09 2020 at 12:22):

I suppose you have looked at systems like "sharding"? But maybe this is offtopic for this stream now...

Eigil Rischel (Jul 09 2020 at 12:23):

I'd love to be part of this reading group, if I can find the time. I grew up in algebraic topology/homotopy theory, so maybe I can help explain stuff like $\Gamma$-spaces.

Fabrizio Genovese (Jul 09 2020 at 12:23):

Yes, I'm looking at sharding too. I mean something easier, actually. The most important thing is to find a nice way to pick a decent $\mathcal{C}^{op}$.

Fabrizio Genovese (Jul 09 2020 at 12:23):

Eigil Rischel said:

I'd love to be part of this reading group, if I can find the time. I grew up in algebraic topology/homotopy theory, so maybe I can help explain stuff like $\Gamma$-spaces.

Great! I suck at organizing things tho. How do we go forward?

Toby Smithe (Jul 09 2020 at 12:23):

Oh, nice @Eigil Rischel -- that would be very interesting!

Toby Smithe (Jul 09 2020 at 12:24):

I too am not very good at organizing things, and am generally overworked as it is but I think it would be worth making a commitment for this one..!

Fabrizio Genovese (Jul 09 2020 at 12:25):

Anyway, @Toby Smithe blockchain and consensus are not as related as you may think. Things are way more subtle there

Fabrizio Genovese (Jul 09 2020 at 12:25):

Both from the theory point of view but most importantly there's not so much interest now because the consensus questions are mainly considered settled and people are focusing on layer 2

Fabrizio Genovese (Jul 09 2020 at 12:25):

Anyway, we can talk about this

Toby Smithe (Jul 09 2020 at 12:25):

Oh man, so much to discuss. Yes, let's have a call soon.

Fabrizio Genovese (Jul 09 2020 at 12:25):

Yups! I'll mail you later then :grinning:

Toby Smithe (Jul 09 2020 at 12:26):

And I don't think I will be able to think much about this Manin-Marcolli paper or a study group on it till at least next week!

Fabrizio Genovese (Jul 09 2020 at 12:27):

Yes, same here! I feel I am not doing anything in particular but then when I look at my agenda I have 3000 things I'm already doing, weirdly

Fabrizio Genovese (Jul 09 2020 at 12:27):

So also for me this should be a sort of low demanding commitment

Toby Smithe (Jul 09 2020 at 12:28):

Yup, same. (... and I should say, really, I don't care so much about "brains" as about "autopoietic" or "cybernetic" systems. Which is one of the reasons I love category theory: it allows me to scratch all these itches.)

Eigil Rischel (Jul 09 2020 at 12:48):

Let me just punt on the organizational question and write a thing here:
For a category $\mathcal{C}$, a $\Gamma$-object of $\mathcal{C}$ is a functor $Fin_* \to \mathcal{C}$, where $Fin_*$ is the category of finite sets with a basepoint. (They're called $\Gamma$-objects because Segal used a category called $\Gamma$, isomorphic to $Fin_*^{op}$, in the paper where he introduced them).

Let's fix some notation: Let $\langle n \rangle$ be the set with $n$ elements and a basepoint, $\{*, 1, \dots, n\}$.
Then there are a family of maps $\rho^i: \langle n \rangle \to \langle 1 \rangle$ which sends the point $i$ to $1$ and everything else to $*$.
A $\Gamma$-object $A$ is special if the map $A(\langle n \rangle) \to A(\langle 1 \rangle)^n$ given by pairing $A(\rho^i)$ for each $i = 1, \dots n$ is an isomorphism.
Now here is a theorem: special $\Gamma$-objects are precisely commutative monoid objects in $\mathcal{C}$ (which I have assumed has finite products).

• Let $m: \langle 2 \rangle \to \langle 1 \rangle$ be the map that sends everything to $1$. Then by composing $A(m)$ with the isomorphism $A(\langle 2 \rangle) \cong A(\langle 1 \rangle)^2$, we get the multiplication map.
• The unit comes from the unique map $\langle 0 \rangle \to \langle 1 \rangle)$

So far so good.
The real trick is if $\mathcal{C}$ "is a homotopy theory" - in this case, I just mean we have some class of "weak equivalences" $W$ in $\mathcal{C}$ that we want to formally invert.
We can form the homotopy category $\mathcal{C}[W^{-1}]$ and ask for commutative monoids in that, but it turns out this notion is very poorly behaved.
Why is it poorly behaved? Basically, asking for a monoid in this category is the same asking for a space with a multiplication and a unit which form a monoid "up to equivalence" - this is like asking for a monoidal category without the coherence axioms.
However, unlike the case of categories, it is not generally true that any "weak monoid", i.e a "monoid up to coherence" can be strictified!
So we'd really like a nice way of encoding the structure of a weak monoid, without having to write down a bunch of coherence conditions.

It turns out special $\Gamma$-spaces do this - if we require that the maps $A(\langle n \rangle) \to A(\langle 1 \rangle)^n$ are weak equivalences, rather than isomorphisms.
This gives "the right" notion of commutative monoid in a homotopy theory.
(For example, given a symmetric monoidal category $\mathcal{C}$, you can build a special-$\Gamma$-category where $\mathcal{C}(\langle 2 \rangle)$ is a category of tuples $(a,b,c, f: a \otimes b \cong c)$, and conversely you can extract a symmetric monoidal category from a special $\Gamma$-category.

Then a $\Gamma$-space is just the instantiation of this stuff in the category of spaces and weak homotopy equivalences.
(Or, more likely, in CW-complexes, which is more convenient for algebraic topology ).

Of course, this is just the basics. The interest in this stuff started because a "grouplike" special $\Gamma$-space, one which "has inverses" (in a certain sense), turns out to be equivalent to an infinite loop space, aka a connective spectrum - a space $X$ equipped with an infinite family of deloopings $X \cong \Omega X_1 \cong \Omega^2 X_2 \dots$ (plus some conditions, namely that all of the deloopings are connected).
This is in turn equivalent to a certain type of homology theory!
But ow we're really getting into the weeds... :grinning:

Aleksandar Makelov (Jul 09 2020 at 12:51):

thanks @Rongmin Lu. Coming from machine learning, I have developed an instinct to doubt things that claim to use heavy mathematics to tell us something about neural networks, but I'm guessing here the focus is more on ways to use interesting math to talk about interesting complex objects in a natural way?

Matteo Capucci (he/him) (Jul 09 2020 at 12:53):

I'd love to be part of the reading group! I didn't have time to properly read the article, but skimming briefly it looked super interesting and a bit over my head in terms of tools.

Eigil Rischel (Jul 09 2020 at 13:04):

Rongmin Lu said:

It doesn't seem to me like they use special $\Gamma$-objects in their work.

I have only skimmed the paper, but I think you're right.
However, my impression was that their $\Gamma$-spaces are all special - they're building them out of symmetric monoidal categories and using them to present spectra . The functor from $\Gamma$-spaces to spectra factors over the "specialization" (the left adjoint to the inclusion of special $\Gamma$-spaces into all of them).

Anyways, the reason I talked so much about special $\Gamma$-spaces is that, at least from my point of view, this is the "point" of $\Gamma$-spaces. Often you work with non-special ones because the full functor category is a lot more convenient, but in some sense the special ones are the "interesting ones".
(To get technical, one works in a model structure on the functor category where the fibrant-cofibrant objects are the special $\Gamma$ spaces valued in Kan complexes).

Fabrizio Genovese (Jul 09 2020 at 14:33):

I think that meeting in a call from time to time would be useful. If I have to present something every now and then I'd keep my tendency to get lazy in check.

Toby Smithe (Jul 09 2020 at 14:38):

Yeah, I think some fairly informal combination of both -- async & occasionally sync -- would be nice.

Sophie Libkind (Jul 09 2020 at 14:58):

I would be interested in a reading group as well! I got to hear Marcolli talk to a group of neuroscientists a couple weeks ago about this work and it was very compelling

Toby Smithe (Jul 09 2020 at 15:20):

Oh, nice! Do you know / can you say which neuroscientists, @Sophie Libkind ?

Sophie Libkind (Jul 09 2020 at 15:46):

It was in Doris Tsao's group meeting. I think they've worked together a bit and taught a class together on the geometry of neuroscience.

xavier (mathematical artist) (Jul 09 2020 at 15:51):

Sophie Libkind said:

It was in Doris Tsao's group meeting. I think they've worked together a bit and taught a class together on the geometry of neuroscience.

Sophie, this link needs authentication to access it.

Brad Theilman (Jul 09 2020 at 16:23):

Hi all, I know some of you from ACT2018. I'd also like to join in a reading group for this paper. My thesis in computational neuroscience starts from the Curto, Itskov, et al. papers they cite (25, 27, 28, 41). I'm recording neural activity from the brains of European starlings and attempting to reconstruct auditory stimulus spaces using those topological methods.

In particular I am interested in identifying what it means for stimuli to be "equivalent" and for neural representations to be "invariant." For example, in our lab we train the birds to recognize two different sounds as indicating that the bird should perform a particular action. What then about the neural activity in the bird's brain signifies that these two physically different sounds "mean" the same thing? The homotopy equivalence idea, that classes of stimuli roughly correspond to homotopy types, is compelling, and we'd like to look for evidence for it in the neural activity.

I would definitely suggest reading reference 27, "Cell groups reveal structure of stimulus space."
I'm also happy to talk more about the experimental neuroscience behind these ideas!

Toby Smithe (Jul 09 2020 at 20:29):

Sophie Libkind said:

It was in Doris Tsao's group meeting. I think they've worked together a bit and taught a class together on the geometry of neuroscience.

Thanks, Sophie! I'm looking forward to digging into this :-)

Tobias Fritz (Jul 09 2020 at 20:34):

Hi all! I've told one of the authors about this channel and the strong interest in the paper (Matilde Marcolli was my PhD advisor). She says that she's currently too busy to follow the discussion, but that it could be very helpful if you pass along any suggestions or comments to her (matilde@caltech.edu)

xavier (mathematical artist) (Jul 09 2020 at 22:02):

Thanks for letting me know. I'll play with my settings and try again.