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Stream: learning: questions

Topic: Walters et. al. Span(RGraph) model

JR Pasmo (Jan 22 2025 at 06:05):

From the learning:questions > Categories applied to interprocess communications discussion I saw,

Chad Nester said:

I really like Span(RGraph) model of Walters et al. (This, for example)

I'm very interested in this model and have a few questions on it, especially now that I see someone really liking it!

First, what makes you (or anyone, if so) really like it, @Chad Nester ? Or if you don't like it, why not?

Then, very basically, what does it mean to be a "reflexive" graph? I might be overthinking it.

Also, I've read on his blog here that they lay out details of the math they used in P. Katis, R.F.C. Walters, The compact closed bicategory of left adjoints, Math. Proc. Camb. Phil. Soc., 130, 77-87, 2001 (here.

I've got TONS of questions about that, but the first, I guess, is: where can I find "modern" papers on what they call "squares"? Are they talking about "adjoint squares" (like in THE Mac Lane p. 103, or "The Double Category of Adjoint squares of Palmqvist" here)? Is the same concept being called something else now? Or has it been updated to something else? I can't find anything conclusive because it's called something so generic and polysemous as "squares"...

@John Baez , does Walters et. al. 2001 paper satisfy the lack of formality you had been missing?

And finally for now, why LEFT adjoints specifically, and not right, for this purpose?

John Baez (Jan 22 2025 at 08:02):

Just one comment for now - it's late here.

Then, very basically, what does it mean to be a "reflexive" graph? I might be overthinking it.

A reflexive graph has a set $V$ of vertices, a set $E$ of edges, two maps $s: E \to V$ and $t: E \to V$, but also a map $i: V \to E$ such that

$s \circ i = t \circ i = 1_V$

In other words, it's a graph where every vertex is equipped with a distinguished edge from that vertex to itself.

By the way, I rarely understand what people mean when they "I may be overthinking it". Usually my undergraduate students said this when they were underthinking something. But here we have a math term, and if one doesn't know the definition one can't understand any sentence containing that term. It's not really a matter of thinking, it's a matter of finding out the definition. So I'm glad you asked!

JR Learnstomath (Jan 26 2025 at 17:51):

John Baez said:

Just one comment for now - it's late here.

Then, very basically, what does it mean to be a "reflexive" graph? I might be overthinking it.

A reflexive graph has a set $V$ of vertices, a set $E$ of edges, two maps $s: E \to V$ and $t: E \to V$, but also a map $i: V \to E$ such that

$s \circ i = t \circ i = 1_V$

In other words, it's a graph where every vertex is equipped with a distinguished edge from that vertex to itself.

By the way, I rarely understand what people mean when they "I may be overthinking it". Usually my undergraduate students said this when they were underthinking something. But here we have a math term, and if one doesn't know the definition one can't understand any sentence containing that term. It's not really a matter of thinking, it's a matter of finding out the definition. So I'm glad you asked!

Thank you not only for the answer (definition) to that question, but also for confirming that just asking the question is really the way to go. The "no question is dumb" slogan of this channel feels true.

David Egolf (Jan 26 2025 at 18:17):

On the topic of reflexive graphs, I think a reflexive graph as described above amounts to a presheaf/functor from (a category that describes how the edges and vertices relate) to (the category of sets). This should mean that the category of reflexive graphs is really nice - it should inherit limits and colimits from the category of sets, for example. In particular, this means you can take pullbacks in this category, which is important for setting up its category of spans.

David Egolf (Jan 26 2025 at 18:17):

I don't know how many of the nice properties of $\mathsf{RGraph}$ are inherited by $\mathrm{Span}(\mathsf{RGraph})$, though. (I'm using $\mathsf{RGraph}$ to refer to the category of reflexive graphs.)

John Baez (Jan 26 2025 at 19:18):

They are very different kinds of categories so you shouldn't expect the latter to directly inherit properties from the former. $\mathsf{RGraph}$ is a category of presheaves, so it's a topos of a particularly simple and nice sort. $\mathbf{Span}(\mathsf{RGraph})$ is a bicategory, and a compact closed symmetric monoidal bicategory. But it's a specially nice one, which Walters and company called a [[cartesian bicategory]].

When we water $\mathbf{Span}(\mathsf{RGraph})$ down to a category $\mathsf{Span}(\mathsf{RGraph})$ by working with isomorphism classes of spans, we get a category that's nothing like a topos. If I remember correctly, binary products in this category are the same as binary coproducts, which is what we expect from linear algebra:

$A \times B \cong A + B$

In a topos, products distribute over coproducts, which is what we expect from set theory:

$A \times (B + C) \cong A \times B + A \times C$

John Baez (Jan 26 2025 at 19:24):

Still, all the exceptionally good properties of $\mathsf{RGraph}$ should transmute into different good properties of $\mathbf{Span}(\mathsf{RGraph})$. This is already at work when we consider why $\mathbf{Span}(\mathsf{RGraph})$ is a cartesian bicategory (a generalization of a 'category of relations'.)

Chad Nester (Feb 03 2025 at 09:36):

@JR Learnstomath: The formalism is very flexible, and I find the notion of system composition present in $\mathsf{Span}(\mathsf{RGraph})$ to be quite convincing. I think a good way to get a feel for this sort of composition is to consider the (admittedly trivial) example of interlocking gears. Section 2.3 of one of my papers gives this example, although I think I first encountered it somwhere else --- Maybe in a talk by Nicoletta Sabadini?

JR Learnstomath (Feb 05 2025 at 09:05):

Oh this is very cool, thanks a lot for sharing!

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