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Stream: learning: questions

Topic: string diagrams = tensor networks?

John van de Wetering (Apr 27 2021 at 17:52):

I was surprised to see that nlab says (https://ncatlab.org/nlab/show/string+diagram): "String diagrams (also Penrose notation or tensor networks)". I thought string diagrams were a much more general concept than tensor networks. A petri net is a string diagram, but not a tensor network right? The other page (https://ncatlab.org/nlab/show/tensor+network) essentially says the same as well.
Btw, thanks @Konstantinos Meichanetzidis for bringing this to my attention

Jules Hedges (Apr 27 2021 at 17:55):

I'd guess that, minor notation differences aside, tensor networks should be something like a string diagrams equipped with a preferred denotation in a category whose morphisms are tensors?

Jules Hedges (Apr 27 2021 at 17:56):

Certainly 99% of the string diagrams I draw have an intended denotation in a category that has nothing to do with tensors

John van de Wetering (Apr 27 2021 at 17:57):

Jules Hedges said:

I'd guess that, minor notation differences aside, tensor networks should be something like a string diagrams equipped with a preferred denotation in a category whose morphisms are tensors?

Yeah that's how I think about it as well. You have your abstract tensor networks, and then a functor into a category of matrices/tensors that counts as your interpretation

John van de Wetering (Apr 27 2021 at 17:58):

Heh, interestingly enough, Wikipedia seems to take the more "pure" approach here: https://en.wikipedia.org/wiki/String_diagram
It only mentions tensors in the references

Jules Hedges (Apr 27 2021 at 17:59):

The article for tensor networks has this key sentence: "In this context, a tensor network is a string diagram with an attitude, in that it is (just) a string diagram (typically regarded in the monoidal category of finite-dimensional vector spaces, but possibly also in super vector spaces etc.)". This seems very confused to me - the denotation in a category of vector spaces is a major piece of additional structure, that's not what "concept with an attitude" means

Jules Hedges (Apr 27 2021 at 18:04):

"A string diagram in a category" is something I'm not sure about. It's something I sometimes say informally, but I'm not sure whether it's something that ought to be said in public. String diagrams are morphisms in the free monoidal category on a monoidal signature, and then the "in a category" part comes from taking a universal functor out of the free monoidal category

John van de Wetering (Apr 27 2021 at 18:08):

Okay, so I think the nlab page should be changed then

Jules Hedges (Apr 27 2021 at 18:12):

Possibly wait for other opinions... there might be a secret good reason why it's written that way...

John van de Wetering (Apr 27 2021 at 18:16):

There's bound to be a wealth of opinions on this topic :)

Konstantinos Meichanetzidis (Apr 27 2021 at 18:24):

Thx for the replies, Jules :)
John also mentioned he suspects Petri nets are not tensor networks. Are they?
In other words, can we just settle the question by counter example?

Amar Hadzihasanovic (Apr 27 2021 at 18:27):

Jules Hedges said:

"A string diagram in a category" is something I'm not sure about. It's something I sometimes say informally, but I'm not sure whether it's something that ought to be said in public. String diagrams are morphisms in the free monoidal category on a monoidal signature, and then the "in a category" part comes from taking a universal functor out of the free monoidal category

I think it's consistent with the usage in non-monoidal category theory that a diagram is a functor, so it is always a “diagram in a category”. You still get the concept that you are talking about by considering “tautological” diagrams given by the identity on the free monoidal category on the signature consisting of that diagram only.

Amar Hadzihasanovic (Apr 27 2021 at 18:27):

(Or its inclusion in a larger free monoidal category, if you prefer.)

John Baez (Apr 27 2021 at 18:49):

String diagrams are much more general than tensor networks: string diagrams work for any symmetric monoidal category (or even more generally), and tensor networks are what you get when you choose the symmetric monoidal category $(\mathsf{Vect}, \otimes)$.

John Baez (Apr 27 2021 at 18:53):

I fixed the nLab page.

Fabrizio Genovese (Apr 27 2021 at 19:56):

Konstantinos Meichanetzidis said:

Thx for the replies, Jules :smile:
John also mentioned he suspects Petri nets are not tensor networks. Are they?
In other words, can we just settle the question by counter example?

Pre-nets would be tensor networks

Fabrizio Genovese (Apr 27 2021 at 19:57):

The main difference between Petri nets and stuff that feels monoidal is that, in general monoidal products do not commute, while in Petri nets transitions have unordered inputs/outputs

John Baez (Apr 27 2021 at 20:04):

In physics, where people talk a lot about tensor networks, "tensor network" means "string diagram in the symmetric monoidal category $(\mathsf{Vect}, \otimes)$". There's a community of people who uses these tensor networks for renormalization in condensed matter physics.

John Baez (Apr 27 2021 at 20:04):

I'd never heard anyone say "tensor network" before papers started coming out of that community.

John Baez (Apr 27 2021 at 20:05):

The nLab has a bunch of links to such papers in the string diagram article.

Fabrizio Genovese (Apr 27 2021 at 21:06):

John Baez said:

In physics, where people talk a lot about tensor networks, "tensor network" means "string diagram in the symmetric monoidal category $(\mathsf{Vect}, \otimes)$". There's a community of people who uses these tensor networks for renormalization in condensed matter physics.

Is this maybe connected with the original intuition by Penrose?

John Baez (Apr 27 2021 at 23:29):

Yes, Penrose was using string diagrams and also his "abstract index notation" primarily to manipulate morphisms in $(\mathsf{Vect}, \otimes)$.

John Baez (Apr 28 2021 at 04:33):

I gave an intro to Penrose's ideas here:

• A prehistory of n-categorical physics.

Nick Hu (Apr 28 2021 at 12:15):

String diagrams don't necessarily have to be confined to monoidal categories either; for a non-monoidal category you can draw string diagrams for morphisms by taking the Poincare dual of pasting diagrams, it's just not particularly interesting because what you get out is essentially a 1D string diagram

Jules Hedges (Apr 28 2021 at 12:17):

Well, string diagrams for monoidal categories are a special case of string diagrams for 2-categories, which have coloured regions

Nick Hu (Apr 28 2021 at 12:18):

Indeed, because a monoidal category is a degenerate 2-category with only one 0-morphism by delooping

Jules Hedges (Apr 28 2021 at 12:20):

Here's me recently learning this fact backwards: https://twitter.com/_julesh_/status/1381601323766980610?s=21

Recently I learned what “delooping” a monoidal category means in category theory, this is previously a word from nLab that I was deeply scared of It means taking a string diagram, and colouring all of the empty regions white

- julesh (@_julesh_)