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Stream: event: Online CT seminar

Topic: June 5 1400 GMT -- Jean-Baptiste Vienney -- Extracting ...

Jean-Baptiste Vienney (May 16 2024 at 21:00):

I'll give a talk on Wednesday June 5. The title is "Extracting an $\mathbb{N}$-graded differential modality from a differential modality".

The previous talk introduced the world of differential categories. This talk is about a recent result related to the first chapter of JS's talk i.e. it is about a new theorem on differential categories.

As was explained a differential category is an additive symmetric monoidal category together with a "differential modality" that we usually denote "$!$". We introduced some time ago with JS the notion of a graded differential category, i.e. an additive symmetric monoidal category with a "graded differential modality" where a smooth map from $A$ to $B$ is axiomatized as a linear map $!_r A \rightarrow B$ where $r \in R$ with $R$ a commutative semiring (i.e. a ring without negatives). Such a map can be thought as a "smooth map of degree r" which is often a polynomial map of degree $r$.

I will explain a theorem which says that under mild conditions one can extract a differential modality graded over $\mathbb{N}$ from a differential modality. This graded modality $(!_{\le n})_{n \in \mathbb{N}}$ is defined such that all the smooth maps from $A$ to $B$ which give $0$ when differentiated $n+1$ times are maps $!_{\le n}A \rightarrow B$. In order to prove this theorem we need a few preliminary results that we will discuss, such as the higher-order Leibniz rule or the Faà di Bruno rule which can be proved in the context of a differential category. We will look at the example of the category $\mathbf{Vec}_{k}^{op}$ together with the symmetric algebra modality. If $k$ is of characteristic $0$ then the extracted modality of degree $n$ provides the polynomials of degree less than $n$ but in positive characteristic the story is more funny as the notion of "differential degree" we're working with doesn't always correspond to the algebraic degree. You can differentiate $f(x)=x^n$ in $\mathbb{Z}_{n}[x]$ to understand the issue!

Jean-Baptiste Vienney (Jun 05 2024 at 00:05):

:bird: : This talk is tomorrow!
:chipmunk: : Thanks for the reminder.

Federica Pasqualone (Jun 05 2024 at 14:21):

Here is the Zoom link for the ongoing meeting: https://us06web.zoom.us/j/82162005589?pwd=GHW8upy3LyerjxQvoGMYV91nVTqFHb.1

Jean-Baptiste Vienney (Jun 05 2024 at 15:10):

Frederica was talking with me about potential use of differential categories in physics. The main thing is maybe that the map $S(A) \otimes A \rightarrow S(A)$ which multiply and the map $S(A) \rightarrow S(A) \otimes A$ which differentiate can be thought as a creation and an annihilation operator respectively.

Jean-Baptiste Vienney (Jun 05 2024 at 15:11):

where $S(A)$ is the symmetric algebra on $A$

Jean-Baptiste Vienney (Jun 05 2024 at 15:12):

By the way it's an $\mathbb{N}$-graded thing. You have $S^n(A) \otimes A \rightarrow S^{n+1}A$ (add a particle) and $S^{n+1}A \rightarrow S^n(A) \otimes A$ (remove a particle).

Eric M Downes (Jun 05 2024 at 15:13):

Thanks Jean-Baptiste!

Slides, video, etc. now in the usual place (see Zoom Link & Dropbox topic in this stream)

Spencer Breiner (Jun 05 2024 at 15:15):

Jean-Baptiste Vienney said:

Frederica was talking with me about potential use of differential categories in physics. The main thing is maybe that the map $S(A) \otimes A \rightarrow S(A)$ which multiply and the map $S(A) \rightarrow S(A) \otimes A$ which differentiate can be thought as a creation and an annihilation operator respectively.

Possibly relevant (search "lowering natural transformation"):
https://arxiv.org/pdf/0706.0711

Federica Pasqualone (Jun 05 2024 at 15:36):

It sounds like a bialgebroid ... hahahahahah

Federica Pasqualone (Jun 05 2024 at 15:39):

btw stay tuned to the n-category theory cafe' this summer guys

David Egolf (Jun 05 2024 at 16:13):

Eric M Downes said:

Thanks Jean-Baptiste!

Slides, video, etc. now in the usual place (see Zoom Link & Dropbox topic in this stream)

I think there might be something wrong with the PDF file from this talk, which I assume contains the slides. When I try to download "Extracting-pres.pdf", my computer tells me the size of the file is 62 bytes. I also can't see any content in the file when I try to open it.

Jean-Baptiste Vienney (Jun 05 2024 at 16:19):

Spencer Breiner said:

Jean-Baptiste Vienney said:

Frederica was talking with me about potential use of differential categories in physics. The main thing is maybe that the map $S(A) \otimes A \rightarrow S(A)$ which multiply and the map $S(A) \rightarrow S(A) \otimes A$ which differentiate can be thought as a creation and an annihilation operator respectively.

Possibly relevant (search "lowering natural transformation"):
https://arxiv.org/pdf/0706.0711

Yes, absolutely!

In this paper they work in the category $\mathbf{FHilb}$. The problem is that FHilb is Not an Interesting (Co)Differential Category in the sense that the only codifferential modality on $\mathbf{FHilb}$ is given by $!A=0$. The issue is that in $\mathbf{FHilb}$ we can't have a lot of differential modality since $!A$ must be finite-dimensional. For instance $! A =S(A)$ cannot work. This problem is solved by graded differential modalities. $\mathbf{FHilb}$ is an $\mathbb{N}$-graded differential modality with $!_n A:=S^n(A)$. But it is not a fully satisfying solution to combine quantum and differential categories because we can't consider coherent states which are element of $S(A)$ but not in any $S^n(A)$, neither in a $\underset{0 \le k \le n}{\bigoplus}S^k(A)$.

Now I think that if we want to have an axiomatic combining dagger compact closed and differentiation, and including coherent states, such that $\mathbf{FHilb}$ would be an example, we should maybe define a notion of relative differential category by replacing the comonad with a relative comonad. In that way the couple $(\mathbf{FHilb},\mathbf{Hilb})$ would maybe form a "dagger compact closed relative codifferential category". In this structure you would consider $A \in \mathbf{FHilb}$ but $!A \in \mathbf{Hilb}$.

I will probably think more about this in the close future and hopefully it can work. :sunglasses:

Jean-Baptiste Vienney (Jun 05 2024 at 16:22):

David Egolf said:

Eric M Downes said:

Thanks Jean-Baptiste!

Slides, video, etc. now in the usual place (see Zoom Link & Dropbox topic in this stream)

I think there might be something wrong with the PDF file from this talk, which I assume contains the slides. When I try to download "Extracting-pres.pdf", my computer tells me the size of the file is 62 bytes. I also can't see any content in the file when I try to open it.

@Eric M Downes ? The file I sent seems to be working so there might be an issue with the uploading on DropBox.

Jean-Baptiste Vienney (Jun 05 2024 at 16:23):

It doesn't work for me either on the DropBox.

Jean-Baptiste Vienney (Jun 05 2024 at 16:28):

Federica Pasqualone said:

It sounds like a bialgebroid ... hahahahahah

I don't know about bialgebroids but differential categories have a version where $!A$ is a bicommutative bialgebra. In this setting you ask for a natural transformation $\overline{d}:A \rightarrow !A$ (the codereliction) and the deriving transformation is recovered like this:
$!A \otimes A \overset{\overline{d} \otimes \mathrm{id}}{\rightarrow} !A \otimes !A \overset{\nabla}{\rightarrow} !A$

Jean-Baptiste Vienney (Jun 05 2024 at 16:29):

This is the version which corresponds to models of differential linear logic.

Federica Pasqualone (Jun 05 2024 at 16:30):

Jean-Baptiste Vienney said:

Spencer Breiner said:

Jean-Baptiste Vienney said:

Frederica was talking with me about potential use of differential categories in physics. The main thing is maybe that the map $S(A) \otimes A \rightarrow S(A)$ which multiply and the map $S(A) \rightarrow S(A) \otimes A$ which differentiate can be thought as a creation and an annihilation operator respectively.

Possibly relevant (search "lowering natural transformation"):
https://arxiv.org/pdf/0706.0711

Yes, absolutely!

In this paper they work in the category $\mathbf{FHilb}$. The problem is that FHilb is Not an Interesting (Co)Differential Category in the sense that the only codifferential modality on $\mathbf{FHilb}$ is given by $!A=0$. The issue is that in $\mathbf{FHilb}$ we can't have a lot of differential modality since $!A$ must be finite-dimensional. For instance $! A =S(A)$ cannot work. This problem is solved by graded differential modalities. $\mathbf{FHilb}$ is an $\mathbb{N}$-graded differential modality with $!_n A:=S^n(A)$. But it is not a fully satisfying solution to combine quantum and differential categories because we can't consider coherent states which are element of $S(A)$ but not in any $S^n(A)$, neither in a $\underset{0 \le k \le n}{\bigoplus}S^k(A)$.

Now I think that if we want to have an axiomatic combining dagger compact closed and differentiation, and including coherent states, such that $\mathbf{FHilb}$ would be an example, we should maybe define a notion of relative differential category by replacing the comonad with a relative comonad. In that way the couple $(\mathbf{FHilb},\mathbf{Hilb})$ would maybe form a "dagger compact closed relative codifferential category". In this structure you would consider $A \in \mathbf{FHilb}$ but $!A \in \mathbf{Hilb}$.

I will probably think more about this in the close future and hopefully it can work. :sunglasses:

well, this is a great idea! @Jean-Baptiste Vienney :sunglasses:

Federica Pasqualone (Jun 05 2024 at 16:30):

Jean-Baptiste Vienney said:

This is the version which corresponds to models of differential linear logic.

I see! Wow, thanks a lot! :sunglasses:

Eric M Downes (Jun 05 2024 at 16:39):

David Egolf said:

Eric M Downes said:

Thanks Jean-Baptiste!

Slides, video, etc. now in the usual place (see Zoom Link & Dropbox topic in this stream)

I think there might be something wrong with the PDF file from this talk, which I assume contains the slides. When I try to download "Extracting-pres.pdf", my computer tells me the size of the file is 62 bytes. I also can't see any content in the file when I try to open it.

Fixed! Thanks for catching that.

David Egolf (Jun 05 2024 at 16:59):

Just skimming the slides, I wanted to mention that this is really cool:

We have an endofunctor $! : L \to L$, the “differential modality”. The idea is that a morphism $!A → B$ $\approx$ a differentiable map from $A$ to $B$.

I quite like the idea of using an endofunctor to define morphisms of a particular kind. It seems simple, but probably powerful!

I was going to say I hadn't seen this kind of thing before, but upon further thought this reminds me of the concept of the Kleisli category for a comonad. But the idea of viewing these morphisms as morphisms "of a particular kind"(e.g. "differentiable") ones, is new to me and I like it! (And this particular example is quite interesting to be aware of as well.)

Jean-Baptiste Vienney (Jun 05 2024 at 17:01):

Indeed, it is very related to the concept of the Kleisli category for a comonad! $!$ is a comonad on $L$ and the Kleisli category is a "cartesian differential category"!

David Egolf (Jun 05 2024 at 17:01):

Most of the slides are rather tough going for me! Conceptually though, I'm curious what the big picture goal is. Is the idea to try and "do calculus" in a really generalized setting? If so, I'm imagining one could try to use (generalized) "derivatives" to do solve certain kinds of optimization problems, maybe?

Or maybe the big picture vision is more related to generalizing certain kinds of logic?

Jean-Baptiste Vienney (Jun 05 2024 at 17:07):

Your two ideas are very good motivations for differential categories! There is a wild notion of calculus which gives cartesian differential categories: functor calculus. A paper which talks about this is Directional derivatives and higher order chain rules for abelian functor calculus. It is the most surprising example of a cartesian differential category in the sense that it is different than usual calculus.

Also differential categories give the semantics of differential linear logic and cartesian differential categories give the semantics of differential lambda calculus. So all this comes from logic.

The idea of graded differential category comes from combining differential linear logic and graded linear logic too.

Jean-Baptiste Vienney (Jun 05 2024 at 17:11):

By the way, I don't claim to understand functor calculus ahah, I just know that it exists. I don't really know much about differential lambda calculus either.

David Egolf (Jun 05 2024 at 17:32):

Interesting stuff! Thanks for explaining!

Federica Pasqualone (Jun 05 2024 at 18:03):

Has anyone come up with a homological algebra flavored paper? :D

Jean-Baptiste Vienney (Jun 05 2024 at 18:04):

I don't think but I've been thinking about something like this:

• work on a symmetric monoidal category $\mathcal{L}$ enriched over abelian groups
• define a graded-commutative algebraic differential modality as an endofunctor $!:\mathcal{L} \rightarrow \mathcal{L}$ with natural transformation $u:A \rightarrow !A$, $\nabla:!A \otimes !A \rightarrow !A$, $\eta:I \rightarrow !A$, $\partial:!A \rightarrow !A \otimes A$ such that each $(!A,\nabla,\eta)$ is a graded-commutative monoid + other equations for $\nabla;\partial$ and $u;\partial$ and $\eta;\partial$.

The exterior algebra functor should be an example. The difference with calculus is that the exterior algebra is not a monad. Also, I don't have ideas of examples other than the exterior algebra.

Jean-Baptiste Vienney (Jun 05 2024 at 18:04):

There should be a version with bialgebra + codereliction also

Jean-Baptiste Vienney (Jun 05 2024 at 18:05):

If you have ideas of other examples, we could write a paper about this :sweat_smile:

Federica Pasqualone (Jun 05 2024 at 18:08):

Homological algebra is always the answer.

David Egolf (Jun 05 2024 at 18:09):

(I don't know if this relates, but I did find this paper: "Forms and exterior differentiation in Cartesian differential categories".)

Jean-Baptiste Vienney (Jun 05 2024 at 18:10):

Oh yes, it relates, you're right. But they don't talk about an endofunctor $!$ like in monoidal differential categories.

Jean-Baptiste Vienney (Jun 05 2024 at 18:11):

I don't think there is yet a notion of an "homological algebra modality".

Federica Pasqualone (Jun 05 2024 at 18:17):

David Egolf said:

(I don't know if this relates, but I did find this paper: "Forms and exterior differentiation in Cartesian differential categories".)

What is this business of a differential site? :heart_eyes: Thanks for the paper!

Jean-Baptiste Vienney (Jun 05 2024 at 18:21):

I have no idea. This is Geoff's business not mine ahah.

Federica Pasqualone (Jun 06 2024 at 13:43):

OK, let me finish the project I am working on. These logical differentials ... I am all in! :sunglasses: