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Stream: learning: questions

Topic: Construction of LagCorel

Nathan McRae (Sep 06 2020 at 23:17):

I'm going through 'A Compositional Framework for Passive Linear Networks', but I'm having trouble with Theorem 7.4 where LagCorel is constructed using Lemma 6.11. My problem is that I can't reconcile the theorem's description of the resulting category with my own direct application of the lemma to $Lag : (Cospan(FinSet), +) \rightarrow (Set, \times)$.

If I understand the lemma correctly, then the morphisms should be (isomorphism classes of) finite corelations $X \rightarrow N \leftarrow Y$ with a decoration in $Lag(N)$ i.e. a Lagrangian relation $\emptyset \nrightarrow \mathbb{F}^{N} \oplus ( \mathbb{F}^{N} )^*$. But the theorem says instead that the morphisms are Lagrangian subspaces of $\mathbb{F}^{X+Y} \oplus ( \mathbb{F}^{X+Y} )^*$.

When I run through the lemma to get composition of decorations, I think what I get makes sense (for decorations n,m I get $S([j_N, j_M]) \circ \lambda ( n, m ) : \emptyset \nrightarrow \mathbb{F}^{\overline{N +_Y M}} \oplus ( \mathbb{F}^{\overline{N +_Y M}} )^*$), but is nothing like the symplectification of the cospan given by the theorem as far as I can see.

Am I not applying the lemma correctly, or am I missing a way to reconcile a direct application of the lemma with the theorem's description of the resulting category?

Cole Comfort (Sep 06 2020 at 23:47):

This is just currying the relation into a state using the compact closed structure of lagrangian relations, and then the decorated cospan framework just gives you a way to compose the states directly. So if you uncurry the state obtained by this, then presumably, you would just get the largrangian subspace of the product space corresponding to composing relations and then currying afterwards.

Nathan McRae (Sep 07 2020 at 02:23):

Thanks for the pointer, I'll see if I can get that through my head

John Baez (Sep 10 2020 at 23:39):

@Nathan McRae - if Cole's answer did not satisfy you, please let me know.

John Baez (Sep 10 2020 at 23:50):

Nathan McRae said:

If I understand the lemma correctly, then the morphisms should be (isomorphism classes of) finite corelations $X \rightarrow N \leftarrow Y$ with a decoration in $Lag(N)$ i.e. a Lagrangian relation $\emptyset \nrightarrow \mathbb{F}^{N} \oplus ( \mathbb{F}^{N} )^*$. But the theorem says instead that the morphisms are Lagrangian subspaces of $\mathbb{F}^{X+Y} \oplus ( \mathbb{F}^{X+Y} )^*$.

There could be several reasons for confusion here, but let me try to eliminate one. A Lagrangian relation $\emptyset \nrightarrow \mathbb{F}^{N} \oplus ( \mathbb{F}^{N} )^*$ is the same thing as a Lagrangian subspace of $\mathbb{F}^{N} \oplus ( \mathbb{F}^{N} )^*$. (That is, there's a natural one-to-one correspondence between these things.)

If you know that, then maybe you're wondering about how we get from stuff involving $\mathbb{F}^N$ to stuff involving $\mathbb{F}^{X+Y}$.

Is that your main concern here?

Nathan McRae (Sep 11 2020 at 02:32):

Yes, thank you, that's exactly my concern. I think I get some of the intuition of what Cole said, but not enough to actually apply it in this situation.

Nathan McRae (Sep 11 2020 at 02:50):

I can see how we could pre-compose a vector in $\mathbb{F}^N$ with $e : X + Y \rightarrow N$ to get a vector in $\mathbb{F}^{X+Y}$. It seems like maybe I should be able to do the same sort of thing with a lagrangian subspace of $\mathbb{F}^N \oplus ( \mathbb{F}^N )^*$, but I don't see exactly how yet.

Is that on the right track, or a mis-step?

John Baez (Sep 11 2020 at 05:31):

This is a complicated paper and it's taking me a while to get back into it. One useful thing, I believe, is Example 6.7. This claims an equivalence between LinRel and LinCorel.

John Baez (Sep 11 2020 at 05:32):

The objects of both categories are finite-dimensional vector spaces. A morphism from V to W in in LinRel is a linear relation from V to W, i.e. a linear subspace of V $\oplus$ W.

John Baez (Sep 11 2020 at 05:34):

A morphism from V to W in LinCorel is a linear corelation from V to W, i.e. an equivalence class of surjective linear maps from V $\oplus$ W to linear spaces S.

John Baez (Sep 11 2020 at 05:36):

The reason this example is important is later we build on it to show LagRel, the category where morphisms are Lagrangian relations between symplectic vector spaces, is equivalent to LagCorel, where morphisms are Lagrangian corelations.

John Baez (Sep 11 2020 at 05:37):

I think understanding this will help us understand your question!

John Baez (Sep 11 2020 at 05:49):

I agree that the statement of Theorem 7.4 is confusing; it's talking about LagCorel but it makes it sound like the morphisms in this category are Lagrangian relations! Of course in Theorem 7.5 we claim LagCorel is equivalent to LagRel, but that comes later.

Cole Comfort (Sep 11 2020 at 05:50):

The reason why LinRel and LinCorel are the same is because they are both constructed as categories of relations over matrices, so in particular, because ${\sf Mat}(k) \cong {\sf Mat}(k)^{\sf op}$ via the transpose functor:

$${\sf LinRel}(k):={\sf Rel} ({\sf Mat}(k))\cong{\sf Rel} ({\sf Mat}(k)^{\sf op}) =: {\sf Corel} ({\sf Mat}(k)) =: {\sf LinCorel}(k)$$

John Baez (Sep 11 2020 at 05:56):

Very nice! I hope Brendan Fong used an explanation like that in his paper "Decorated corelations" - I'll have to check. In our paper we seem to claim this equivalence without giving a good explanation!

I think part of the problem is that we were getting pretty tired by the time we finished this paper - we rewrote it after getting a referee's report, and everything about decorated corelations was added at that time.

John Baez (Sep 11 2020 at 05:57):

Brendan knew all this stuff cold at that point, so he wrote very terse explanations, and my job was to expand and clarify them.

John Baez (Sep 11 2020 at 05:58):

I don't know if I'll have the energy to rewrite the arXiv version, but right now you're making me want to!

John Baez (Sep 11 2020 at 05:58):

At that time Brendan was actually half-wanting to write a whole book about this stuff....

Cole Comfort (Sep 11 2020 at 05:59):

Yes Brendan Fong's thesis pairs well with Fabio Zanasi's thesis which explains these things in maybe more detail... and probably wine and cheese as well.

Cole Comfort (Sep 11 2020 at 06:08):

It would have been nice if there were more string diagrams in the paper for Lagrangian relations for example, because they turn out to be quite pretty and oddly symmetrical when they are fleshed out in all their glory. The symplectification functor from linear relations has a particularly nice graphical interpretation in terms of doubling which looks quite elegant.

John Baez (Sep 11 2020 at 06:26):

"The paper for Lagrangian relations" - you mean the paper I wrote with Brendan? Earlier versions of this on the arXiv have more string diagrams.

Cole Comfort (Sep 11 2020 at 06:30):

Yes, the paper you wrote with brendan. Maybe I should check them out!

Cole Comfort (Sep 11 2020 at 06:33):

Hopefuly you guys didn't already work through everything that I thought we had discovered!

John Baez (Sep 11 2020 at 06:36):

Hopefully we did but you still write a paper about it! :upside_down:

John Baez (Sep 11 2020 at 06:37):

This stuff needs to be explained better, and there are lots of different perspectives one could take on it, so don't let anything we did prevent you from writing more!

John Baez (Sep 11 2020 at 06:43):

By the way, if you want stuff on string diagrams and Lagrangian relations, try these papers if you haven't already:

• John Baez and Brandon Coya, Props in network theory.

• Brandon Coya, A compositional framework for bond graphs.

John Baez (Sep 11 2020 at 06:44):

They're combined here:

• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, PhD thesis, U.C. Riverside, 2018.

John Baez (Sep 11 2020 at 06:46):

There's a really nice story here....

Nathan McRae (Sep 11 2020 at 17:30):

John Baez said:

One useful thing, I believe, is Example 6.7. This claims an equivalence between LinRel and LinCorel.

Well then it makes sense that I'm struggling with Theorem 7.4, because right before this I was struggling with Example 6.7!

I did initially check out the paper it referenced (http://www.tac.mta.ca/tac/volumes/33/22/33-22abs.html, section 7.2), but it went over my head at the time and I just tried going at it myself. I fumbled around until I ended up trying to show the equivalence via pushouts of linear relations and pullbacks of linear corelations but I got stuck. Looking at the referenced paper again, I think I can now follow it better and and see where I went wrong.

Nathan McRae (Sep 11 2020 at 17:30):

Cole Comfort said:

The reason why LinRel and LinCorel are the same is because they are both constructed as categories of relations over matrices, so in particular, because ${\sf Mat}(k) \cong {\sf Mat}(k)^{\sf op}$ via the transpose functor:

$${\sf LinRel}(k):={\sf Rel} ({\sf Mat}(k))\cong{\sf Rel} ({\sf Mat}(k)^{\sf op}) =: {\sf Corel} ({\sf Mat}(k)) =: {\sf LinCorel}(k)$$

Cole, I like the elegance of your explanation, but I'm afraid I'm not yet mature enough to get it via that alone. I think I'll need to come at it from the other direction: I'll have to wade through the lower-level details, then use those to grok your explanation.

Nathan McRae (Sep 11 2020 at 17:38):

For my own part I'm very interested to explore these, and so having a paper which takes the specific example of linear passive circuits and runs with it is extremely helpful as a jumping-off point.

John Baez (Sep 11 2020 at 18:01):

Great!

Cole Comfort (Sep 11 2020 at 18:56):

@Nathan McRae I would suggest that you also read the blog graphicallinearalgebra.net because it gives gives a construction of linear relations in terms of string diagrams starting from basic arithmetic using string diagrams. In this notation, the transpose looks like flipping diagrams horizontally. And the orthogonal complement looks like swapping colours; where the axioms of their presentation of linear relations is invariant under colour change and horizontal symmetry.

Nathan McRae (Sep 11 2020 at 19:52):

Thanks Cole, by luck I added that to my bookmarks the other day while exploring something unrelated, so I'm glad to find that it's more pertinent than I first thought.

Cole Comfort (Sep 11 2020 at 20:51):

Yes, and the symplectification functor corresponds to doubling a linear relation and changing the colours in one of the copies of the original linear relation.