## Network Theory (Part 31)

Last time we came up with a category of labelled graphs and described circuits as ‘cospans’ in this category.

Cospans may sound scary, but they’re not. A cospan is just a diagram consisting of an object with two morphisms going into it:

We can talk about cospans in any category. A cospan is an abstract way of thinking about a ‘chunk of stuff’ $\Gamma$ with two ‘ends’ $I$ and $O.$ It could be any sort of stuff: a set, a graph, an electrical circuit, a network of any kind, or even a piece of matter (in some mathematical theory of matter).

We call the object $\Gamma$ the apex of the cospan and call the morphisms $i: I \to \Gamma, o : O \to \Gamma$ the legs of the cospan. We sometimes call the objects $I$ and $O$ the feet of the cospan. We call $I$ the input and $O$ the output. We say the cospan goes from $I$ to $O,$ though the direction is just a convention: we can flip a cospan and get a cospan going the other way!

If you’re wondering about the name ‘cospan’, it’s because a span is a diagram like this:

Since a ‘span’ is another name for a bridge, and this looks like a bridge from $I$ to $O,$ category theorists called it a span! And category theorists use the prefix ‘co-‘ when they turn all the arrows around. Spans came first historically, and we will use those too at times. But now let’s think about how to compose cospans.

Composing cospans is supposed to be like gluing together chunks of stuff by attaching the output of the first to the input of the second. So, we say two cospans are composable if the output of the first equals the input of the second, like this:

We then compose them by forming a new cospan going all the way from $X$ to $Z$:

The new object $\Gamma +_Y \Gamma'$ and the new morphisms $i'', o''$ are built using a process called a ‘pushout’ which I’ll explain in a minute. The result is cospan from $X$ to $Z,$ called the composite of the cospans we started with. Here it is:

So how does a pushout work? It’s a general construction that you can define in any category, though it only exists if the category is somewhat nice. (Ours always will be.) You start with a diagram like this:

and you want to get a commuting diamond like this:

which is in some sense ‘the best’ given the diagram we started with. For example, suppose we’re in the category of sets and $Y$ is a set included in both $\Gamma$ and $\Gamma'.$ Then we’d like $A$ to be the union of $\Gamma$ and $\Gamma.$ There are other choices of $A$ that would give a commuting diamond, but the union is the best. Something similar is happening when we compose circuits, but instead of the category of sets we’re using the category of labelled graphs we discussed last time.

How do we make precise the idea that $A$ is ‘the best’? We consider any other potential solution to this problem, that is, some other commuting diamond:

Then $A$ is ‘the best’ if there exists a unique morphism $q$ from $A$ to the ‘competitor’ $Q$ making the whole combined diagram commute:

This property is called a universal property: instead of saying that $A$ is the ‘best’, grownups say it is universal.

When $A$ has this universal property we call it the pushout of the original diagram, and we may write it as $\Gamma +_Y \Gamma'.$ Actually we should call the whole diagram

the pushout, or a pushout square, because the morphisms $i'', o''$ matter too. The universal property is not really a property just of $A,$ but of the whole pushout square. But often we’ll be sloppy and call just the object $A$ the pushout.

Puzzle 1. Suppose we have a diagram in the category of sets

where $Y = \Gamma \cap \Gamma'$ and the maps $i, o'$ are the inclusions of this intersection in the sets $\Gamma$ and $\Gamma'.$ Prove that $A = \Gamma \cup \Gamma'$ is the pushout, or more precisely the diagram

is a pushout square, where $i'', o''$ are the inclusions of $\Gamma$ and $\Gamma$ in the union $A = \Gamma \cup \Gamma'.$

More generally, a pushout in the category of sets is a way of gluing together sets $\Gamma$ and $\Gamma'$ with some ‘overlap’ given by the maps

And this works for labelled graphs, too!

Puzzle 2. Suppose we have two circuits of resistors that are composable, like this:

and this:

These give cospans in the category $L\mathrm{Graph}$ where

$L = (0,\infty)$

(Remember from last time that $L\mathrm{Graph}$ is the category of graphs with edges labelled by elements of some set $L.$) Show that if we compose these cospans we get a cospan corresponding to this circuit:

If you’re a mathematician you might find it easier to solve this kind of problem in general, which requires pondering how pushouts work in $L\mathrm{Graph}.$ Alternatively, you might find it easier to think about this particular example: then you can just check that the answer we want has the desired property of a pushout!

If this stuff seems complicated, well, just know that category theory is a very general, powerful tool and I’m teaching you just the microscopic fragment of it that we need right now. Category theory ultimately seems very simple: I can’t really think of any math that’s simpler! It only seem complicated when it’s unfamiliar and you have a fragmentary view of it.

So where are we? We know that circuits made of resistors are a special case of cospans. We know how to compose cospans. So, we know how to compose circuits… and in the last puzzle, we saw this does just what we want.

The advantage of this rather highbrow approach is that a huge amount is known about composing cospans! In particular, suppose we have any category $C$ where pushouts exist: that is, where we can always complete any diagram like this:

to a pushout square. Then we can form a category $\mathrm{Cospan}(C)$ where:

• an object is an object of $C$

• a morphism from an object $I \in C$ to an object $O \in C$ is an equivalence classes of cospans from $I$ to $O:$

• we compose cospans in the manner just described.

Why did I say ‘equivalence class’? It’s because the pushout is not usually unique. It’s unique only up to isomorphism. So, composing cospans would be ill-defined unless we work with some kind of equivalence class of cospans.

To be precise, suppose we have two cospans from $I$ to $O$:

Then a map of cospans from one to the other is a commuting diagram like this:

We say that this is an isomorphism of cospans if $f$ is an isomorphism.

This gives our equivalence relation on cospans! It’s an old famous theorem in category theory—so famous that it’s hard to find a reference for the proof—that whenever $C$ is a category with pushouts, there’s a category $\mathrm{Cospan}(C)$ where:

• an object is an object of $C$

• a morphism from an object $I \in C$ to an object $O \in C$ is an isomorphism class of cospans from $I$ to $O.$

• we compose isomorphism classes of cospans by picking representatives, composing them and then taking the isomorphism class.

This takes some work to prove, but it’s true, so this is how we get our category of circuits!

Next time we’ll do something with this category. Namely, we’ll cook up a category of ‘behaviors’. The behavior of a circuit made of resistors just says which currents and potentials its terminals can have. If we put a circuit in a metaphorical ‘black box’ and refuse to peek inside, all we can see is its behavior.

Then we’ll cook up a functor from the category of circuits to the category of behaviors. We’ll call this the ‘black box functor’. Saying that it’s a functor mainly means that

$\blacksquare(f g) = \blacksquare(f) \blacksquare(g)$

Here $f$ and $g$ are circuits that we can compose, and $f g$ is their composite. The black square is the black box functor, so $\blacksquare(fg)$ is the behavior of the circuit $f g.$ There’s a way to compose behaviors, too, and the equation above says that the behavior of the composite circuit is the composite of their behaviors!

This is very important, because it says we can figure out what a big circuit does if we know what its pieces do. And this is one of the grand themes of network theory: understanding big complicated networks by understanding their pieces. We may not always be able to do this, in practice! But it’s something we’re always concerned with.

### 37 Responses to Network Theory (Part 31)

1. Bruce Smith says:

Wikipedia says morphisms in the category of sets are (in general) functions, not just inclusions, in which case the pushout is not as simple as the union. Obviously you know this — mainly I’m just mentioning it as a clarification. But I guess there is also a question, since it seems to me that in the pure category of sets there is no canonical way to understand certain morphisms as being “inclusions” (except for their one->one as opposed to many->one nature), so even to define things like “union” you probably need something as fancy as the pushout. Is that right?

• John Baez says:

Wikipedia says morphisms in the category of sets are (in general) functions, not just inclusions, in which case the pushout is not as simple as the union.

Right. In the lecture, starting around minute 22, I explained the general recipe for doing pushouts in the category of sets, and did an example where the morphisms aren’t inclusions.

But I guess there is also a question, since it seems to me that in the pure category of sets there is no canonical way to understand certain morphisms as being “inclusions”…

Actually there is! In any category there’s a concept of a morphism being ‘monic’. In the category of sets this reduces to a function being one-to-one, and in other categories it’s usually the right generalization of being one-to-one.

Like the category of sets, the category of graphs and the category of labelled graphs have some nice properties involving monics and pushouts. For these nice categories (called ‘adhesive’)

whenever both $o$ and $i'$ are monic, the pushout exists. Moreover, whenever $o$ is monic and the pushout exists, then $o''$ is monic. By symmetry, a similar thing holds for $i'$ and $i''.$

So, for these nice categories, when both $o$ and $i'$ are monic, the pushout exists and everything is monic, so the situation closely resembles the situation in the category of sets when we’re taking the union of two sets $\Gamma$ and $\Gamma'$ whose intersection is $Y.$

However, you are right that in a general category, it’s best to start with the general concept of pushout and then consider special kinds, instead of trying to start with some concept of ‘union’.

The problem with traditional set theory is that given two sets we expect them to ‘know ahead of time’ what their ‘overlap’ is: which elements they have in common. In category theory we take a different view; we provide this information via the pushout data:

In set theory there’s something called the ‘disjoint union’ where you force two sets to be disjoint before taking their union, by coloring the elements of one red and the other green, or some trick like that. This generalizes nicely to other categories: it’s the coproduct, and it’s significantly simpler than the pushout. If you watch the lecture you’ll see how I build a pushout in the category of sets starting with a coproduct. This trick generalizes, with a bit of work.

As you can see, there’s a rather long process of relearning math so you can do everything in a general category, or at least a general ‘sufficiently nice’ category, instead of just in the category of sets. A true category theorist is someone who has completely mastered this way of thinking. I’m still an amateur, but it’s a nice outlook to have.

• Eugene says:

John, if you have been thinking of category theory for something like 15-20 years longer than me, and you are still an amateur, what does it take to become a master? I weep in frustration.

• John Baez says:

I’ve been thinking about category theory for less than 1/15 of my waking hours for the last 15 years, so not even one year.

I spent a lot of the last 15 years trying to understand n-categories and how to apply them to mathematical physics. But that is very different than what I mean by category theory here: namely, the game of all taking all theorems of set-based mathematics and generalizing them to suitably nice categories. I only picked up a bit of that stuff on the side. The people who are good at that stuff tend to have studied category theory in grad school. These are the people who know the difference between adhesive and extensive categories, remember the definition of a regular epimorphism, know several variants of the Special Adjoint Functor Theorem, know exactly when the axiom of choice is required to prove any basic result in math, and know how to get around that when you’re in a non-Boolean topos.

Luckily, I have lots of friends like this: Tom Leinster, Todd Trimble, Steve Lack, Mike Shulman, etc. So, when I need information of this sort I can cheat and ask them. When more god-like powers are required, I summon up Ross Street and André Joyal. But I don’t feel I’ve ever proved a nontrivial result in category theory… and I’ve rarely even used any nontrivial results. What I’ve mainly done is apply ideas from category theory or n-category theory to subjects where people hadn’t been doing that yet. Get in, make the kill, and grab some meat before the lions arrive!

I feel quite comfortable with the categorical way of thinking, so in that sense I’m not an amateur. But I can’t remember many theorems about category theory, and I feel I have very little ‘technique’: I don’t have ‘tricks up my sleeve’. I don’t expect that to change. So in that sense I’m a hardened amateur.

• Todd Trimble says:

I really feel your old pal James Dolan deserves a mention!

• John Baez says:

Perhaps I should have mentioned him, but I deliberately didn’t, because while he taught me most of the category theory I know, and completely transformed my understanding of mathematics, he’s not the kind of person whom I’d turn to if suddenly I wanted to know whether pushouts preserve monics in a regular category… whereas you are. It’s because I learned most of my category theorem from him that I feel I’m an amateur in category theory: I’ve got a lot of the right intuitions, but very few ‘tricks’ or ‘techniques’. I try to make up for this by having weird new ideas about what to do with category theory. Then I turn to people like you to help me actually get things to work, unless I can just hack it out myself. (Not that there’s anyone ‘like you’.)

• Bruce Smith says:

And where/how did James learn category theory?

• Todd Trimble says:

I see what you mean. James, for all his terrific intuition (which has been transformative for me as well), is not someone I think of as technique-driven like some of those other names. (James strikes me as largely self-taught in category theory, but he did spend some time at SUNY Buffalo which is where Lawvere and Schanuel are, and surely that helped.) Incidentally — not that you’re seriously asking — but pushouts of monics in regular categories need not be monic. A good example is in the category of commutative rings, where the pushout of the inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ along $\mathbb{Z} \to \mathbb{Z}/(2)$ is not monic. :-)

• John Baez says:

Todd wrote:

Incidentally — not that you’re seriously asking — but pushouts of monics in regular categories need not be monic. A good example is in the category of commutative rings, where the pushout of the inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ along $\mathbb{Z} \to \mathbb{Z}/(2)$ is not monic. :-)

Wow! So now everyone can see what I meant: when I’ve got some question like this, I know whom to turn to.

2. Blake Stacey says:

Typo: missing “latex” before “\blacksquare{fg}”.

3. Bruce Smith says:

This might be digressing more than you’d like (so feel free to push this discussion somewhere else), but —

With matrices (or linear maps/operators), we have not only vector operations, but a couple kinds of products (matrix multiplication, tensor/kronecker product), many norms, even exponentiation (e^X for a matrix X) — and these have many interrelations (e.g. both kinds of products are bilinear in their inputs, matrix multiplication also sort-of-distributes over tensored inputs in a nice way (“distributes” is the wrong term but I think you can guess what I refer to), many norms are submultiplicative…).

Is all that able to be studied more abstractly in category theory, and is doing so worthwhile? (E.g. I have never heard a definition of what “exponentiation” should mean in a general category, but I’m guessing it’s been done.) A motivation, as always, might be to understand it more clearly, guess at missing or suboptimally expressed theorems, or help think up generalizations.

• John Baez says:

Very briefly, most or all of this stuff has been studied more abstractly in category theory, so we don’t have to keep reproving the same theorems over and over. For example, you’re talking about things we can do in the category $\mathrm{FinVect}$ of finite-dimensional vector spaces… but you can also do most of these things in the category of finite-dimensional representations of a group, which is a very similar category: in fact it’s the category of functors from $G$ to $\mathrm{FinVect},$ where we think of $G$ as a category with one object. We can also do many of these things in the category of finite-dimensional vector bundles over a space. And mathematicians and physicists really need these generalizations, and a lot more!

The key is to find a bunch of useful properties of categories: being monoidal, braided, symmetric, complete, cocomplete, closed, enriched over some other category, etc. These properties count as useful if they tend to show up a lot ‘in nature’ and if familiar constructions and theorems work in any category with some conjunction of these properties. Category theorists have been working pretty hard since around 1950 trying to find these properties and generalize mathematics in this way. So, by now there’s quite a lot to say about which of the constructions you mention generalize to which kinds of categories.

Borceux’s three-volume Handbook of Categorical Algebra and Johnstone’s partially completed three-volume Sketches of an Elephant are good places to tour this kind of work. The table of contents of the latter is probably enough to crush your interest.

• Bruce Smith says:

That table of contents makes it clear there’s a lot there, but gives me no clue how to look up a conventional concept (e.g. Kronecker product, Shannon entropy, etc) to find out whether/how it’s been treated in category theory. And the Handbook of Categorical Algebra doesn’t appear to be available online….

• John Baez says:

Unfortunately I don’t know a rapid way to find out how some particular topic has been treated using category theory, except by asking a good category theorist. The nLab is quite good, but the coverage is still spotty, simply because there’s so much material that should in principle be in there.

You can’t really take your favorite everyday mathematical concept and look it up in the The Elephant, unless it’s a very fundamental concept like ‘and’, ‘there exists’, ‘implies’, ‘power set’, ‘natural numbers’, or ‘real numbers’. The Elephant is mainly about a hierarchy of better and better kinds of categories leading up to the ‘best categories of all’: toposes, in which you can do all of logic, set theory, analysis, etc. in a generalized sort of way. When I say ‘best’ I just mean ‘a lot like the category of sets, but without assuming the principle of excluded middle’. The category of graphs is a nice example of a topos. En route to these ‘best’ categories Johnstone proves a lot of theorems about various kinds of ‘good, but not quite that good’ categories. As he ascends the ladder, more and more basic concepts of mathematics kick in. But for a tour of concepts related to linear algebra you’d need a very different book on category theory.

• Bruce Smith says:

Unfortunately I don’t know a rapid way to find out how some particular topic has been treated using category theory, except by asking a good category theorist.

If someone who knows category theory well wants to popularize its use, maybe they should add to each Wikipedia entry on X, a comment (with references) about how X has been treated in category theory. (But only when this treatment is mature enough to perhaps be useful outside of category theory.)

That is probably more effective than just producing a good centralized document specific to category theory (like a specialized wiki, or something analogous to the Complexity Zoo), since someone wanting to understand X will be more likely to check Wikipedia anyway, and might not even know about the centralized document.

But for me, the centralized document would be fine if I knew about it… so I’ll browse the nLab.

• John Baez says:

In case anyone is wondering, the nLab was started by members of the group blog The n-Category Café. This blog started with the philosopher David Corfield, the mathematical physicists Urs Schreiber and myself. It now has many more authors. The general theme is using categories and n-categories to revolutionize mathematics and physics. At some point we decided to collect this information in a more systematic way on a wiki, and thus the nLab was born.

Unfortunately this happened right around the time I got involved in environmental issues and lost my taste for highly theoretical physics and pure math, so I never contributed much to this wiki: instead, I mostly quit posting on the The n-Category Caé and started this blog here. Now I’m trying to use categories and n-categories to revolutionize applied mathematics, engineering and biology. But I want to keep the fancy math to a bare minimum, so I don’t scare away the people I want to talk to!

Right now the people contributing heavily to the nLab seem a bit disjoint from those who blog most on the The n-Category Caé. The nLab crowd mainly communicates on the nForum.

So, people who want to learn how categories are used in various subjects need to poke around a bit, but there’s a huge amount to be found online.

4. Eugene says:

John wrote:

Unfortunately I don’t know a rapid way to find out how some particular topic has been treated using category theory, except by asking a good category theorist.

I will take it as an invitation to ask a question here: do undirected multigraphs with loops form a topos?

• John Baez says:

This is the sort of thing Todd could answer in a minute. What I call graphs, namely directed multigraphs allowing loops, form the category of presheaves on

$\bullet \stackrel{\longrightarrow}{\longrightarrow} \bullet$

so that’s why they form a topos. I haven’t thought about the undirected ones nearly as much, so I can’t instantly spit out the answer. Not having any tricks up my sleeves, I’d try to see if there’s an internal hom and a subjobject classifier. The subobject classifier for what I call graphs is very appealing:

• Sebastiano Vigna, A guided tour in the topos of graphs.

This might or might not help:

• Bill Lawvere (1989), Qualitative distinctions between some toposes of generalized graphs, in Categories in Computer Science and Logic (Boulder, CO, 1987), volume 92 of Contemporary Mathematics, American Mathematical Society, Providence, RI, 261–299.

• Eugene says:

Yes, I was hoping Todd would.

I know about directed graphs being a presheaf hence a topos and of Vigna’s paper (which is best read after you learn a definition of a topos). I had a look at Lawere’s paper. It seems to be about something quite different. But R. Brown, I. Morris, J. Shrimpton and C.D. Wensley in Graphs of morphisms of graphs (The Electronic Journal of Combinatorics 15 (2008)) do seem to say that undirected graphs form a topos.

They think of undirected graphs as graphs with involutions, which is almost but not quite how you are thinking of your graphs: you don’t require that maps of graphs to preserve involutions…

• Todd Trimble says:

Unfortunately, I can’t answer in a minute! Well, I can answer with a gut reaction (that it isn’t a topos) in a second, but I’d need more time than a minute to justify that, and I could be wrong.

It could be a problem with terminology. By a graph with involution those authors mean, I would guess, a functor $C \to Set$ where $C$ is the category with two objects $1, 0$ and three nonidentity morphisms $s, t: 1 \to 0$ and $\sigma: 1 \to 1$ such that $\sigma^2 = 1$ and $t \sigma = s$. (Or do they mean something else?) Certainly the category of such functors is an example of a (presheaf) topos. Perhaps I’m not thinking clearly, but I’m not seeing why that would be equivalent to an undirected multigraph with loops the way I usually think of it, given by a triple $(E, V, f: E \to \binom{V}{2})$ where the codomain of $f$is the quotient of $V \times V$ by the $\mathbb{Z}/(2)$-action. To produce a functor $C \to Set$ from such data, presumably one considers exactly two edges (related by the involution) for each undirected edge, which means we’d be ruling out fixed points of the involution from consideration. Suffice it to say that the sticking point for me are the possible ramifications of fixed points of the involution.

Sorry, I need more time. Maybe ask another category theorist? :-)

• John Baez says:

I think you’re doing great, Todd. I agree, the ordinary kind of undirected multigraph with loops is not the same as a graph with an involution as you define it (which is the reasonable way to define it).

Consider the case where we have just one vertex and two edges (both of which are necessarily loops). There’s just one isomorphism class of undirected multigraphs with loops of this sort. But there are two isomorphism classes of graphs with involution of this sort: one where the involution switches the two edges, and one where it doesn’t.

• John Baez says:

I think Todd’s remarks basically agree with this paper:

• R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, Graphs of morphisms of graphs, The Electronic Journal of Combinatorics 15 (2008).

This paper answers Eugene’s question, too: it says the usual category of undirected multigraphs with loops is not a topos. Well, they actually say:

The question of what should be an undirected graph, and morphisms of such graphs, leads to the question of the properties of the corresponding category. It turns out that the most transparent and simple definition does not yield a cartesian closed category. This makes it difficult to decide what should be (if at all) the automorphism object AUT(G) of such an undirected graph G, since we cannot then ensure that AUT(G) has the structure of both an undirected graph and a group. In particular there is not a clear notion of composition AUT(G) × AUT(G) → AUT(G).

If it ain’t cartesian closed, it ain’t a topos. Then they vow to get around this by making up some other concept of undirected graph:

There is however an alternative definition of undirected graphs and their morphisms, namely to consider an undirected graph as a directed graph with a reversal. This is the approach taken by Bumby and Latch in [7] and in several other sources, and does yield a topos $\mathrm{Rdgrph}.$

This ‘reversal’ is what Todd was calling an involution. However, there’s one slight difference between their concept and Todd’s: before slapping on the involution, they start with a reflexive directed multigraph with loops, meaning one where each vertex has a distinguished loop. (Think: just like each object in a category has an identity morphism.)

Then, when they slap on the involution, require that this distinguished loop be fixed by the involution. (Think: just like any identity morphism in a groupoid is its own inverse.)

All these conditions are nice and simple and equational, so $\mathrm{Rdgraph}$ is a presheaf topos.

There’s also a topos of directed multigraphs with loops (which in my seminar I just call graphs), and one of reflexive directed multigraphs with loops. Both these are presheaf topoi too. The authors say “The distinction between the two toposes is eloquently argued by Lawvere”—in the paper I cited earlier in this conversation.

• Todd Trimble says:

Some additional facts may be of interest to people reading this thread. Suppose we consider instead the “simple” (i.e., non-multi) version of undirected graph. This can be identified with a set equipped with a symmetric relation (and similarly a reflexive simple undirected graph can be identified with a set equipped with a reflexive symmetric relation). In both cases, the category of such simple graphs isn’t a topos, but it’s almost as nice as a topos, being what is called a quasitopos. In particular it’s cartesian closed (even better, it’s locally cartesian closed, meaning all its slices are cartesian closed), and has many of the nice exactness properties of the category of sets. [In case anyone feels like pursuing this, these quasitoposes arise as separated presheaves for a suitable Grothendieck topology on a corresponding presheaf topos of graphs like we were discussing above. In fact, the Grothendieck topology is the $\neg\neg$-topology, which you can read about in the topos theory book by Mac Lane and Moerdijk.]

• John Baez says:

Cool! One upon a time Alex Hoffnung and I proved that simplicial complexes form a quasitopos. Again, it’s because they’re separated presheaves. James Dolan was the one who suggested proving this, and we did it while proving a bunch of other things were quasitoposes. Since Alex and I were amateurs in this field, our paper should be fairly easy to read even for people who know nothing of toposes, much less quasitoposes.

Anyway, I sense a strong family resemblance between simplicial complexes and simple graphs: I guess the latter can be seen as the 1-dimensional ‘truncation’ of the former. (Somehow the product of two simple graphs is still 1-dimensional in this truncation, while in the category of simplicial sets it would be 2-dimensional. But this reminds me of other things that work this way, like the product of categories.)

5. […] Remember, circuits made of resistors are cospans. This lets us talk about isomorphisms between them. If you forget the how isomorphism between cospans work, you can review it in Part 31. […]

6. Eugene says:

I am very glad you and Alex Hoffnung wrote that paper. It’s very useful to have a categorical point of view of diffeological spaces particularly since there is very little category theory in Iglesias-Zemmour’s book.

My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally. Lee and I played around with them a bit but got nowhere. It may matter that they form a quasitopos. Perhaps we can talk about it next May.

• John Baez says:

Yay, finally someone says they like that paper! People who don’t like category theory are happy not knowing that diffeological spaces are concrete sheaves. People who do often want to drop the ‘concreteness’ condition and work simply with sheaves, thus obtaining a topos instead of a mere quasitopos. (This means generalizing diffeological spaces to things that don’t ‘have enough points’: e.g. you could get a thing with lots of smooth curves in it but only one point.) It’s nice to finally meet a reader who likes category theory but not too much.

My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally.

Sure, let’s talk about this! But a clue now would be nice, if it’s not top secret.

• Tobias Fritz says:

My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally.

Let me make a wild guess of how this goes!

Actual physical systems are often modelled as networks of interacting subsystems. If you draw each subsystem as a vertex and put a simplex for every collection of subsystems which interacts, then you get a simplicial complex! More precisely, the total Hamiltonian is a sum of interaction terms like this:

$H = \sum_{S\in X} H_S.$

Here, $X$ is the simplicial complex describing which interactions there are, and $S$ ranges over all its simplices. If $S$ is a simplex, then any subset $T\subseteq S$ should be a simplex as well: if all the subsystems forming $S$ interact, the subsystems making up $T$ automatically interact as well, and this is why you get an (abstract) simplicial complex.

Alternatively, you can only sum over all maximal simplices $S$ and attribute each “smaller” interaction term to an arbitrary maximal simplex which contains it. (But for many Hamiltonians that would be quite an odd thing to do! Think e.g. of $H =$ kinetic terms + pairwise interaction terms.)

In the case that the simplicial complex is a clique complex, this is closely related to Markov networks and the Hammersley-Clifford theorem.

So this is at least one way in which simplicial complexes come up in Hamiltonian systems — Eugene, is this what you have in mind? I wonder how it relates to electrical circuits?

• John Baez says:

That’s a nice guess, Tobias… but I’m a bit suspicious of whether this is what Eugene meant, because the same idea would work for arbitrary dynamical systems if we summed the vector fields generating time evolution, instead of the Hamiltonians. Since Eugene knows and loves networks of general dynamical systems, not just Hamiltonian systems, why would he have said

My second comment is that when you are dealing with networks of Hamiltonian (a.k.a. conservative) systems, simplicial complexes come up naturally.

thus unnecessarily limiting the generality of the idea you just mentioned? Except perhaps to deliberately confuse us.

• Tobias Fritz says:

Ah, yes. So then I can’t wait to hear his answer ;)

• Eugene says:

thus unnecessarily limiting the generality of the idea you just mentioned?

I think I am confused and probably accidentally confused you.

When one says “simplicial complex”, are the vertices necessarily ordered? What do you call an unordered simplicial complex? Or, better yet, maybe for Hamiltonian systems one needs symmetric simplicial sets?

• John Baez says:

Eugene wrote:

When one says “simplicial complex”, are the vertices necessarily ordered?

I don’t know. Some people say a simplicial complex is a simplicial set with an extra property; with this sort of definition, the vertices in any simplex are ordered. This is what you’d want in algebraic topology: you want a simplex to determine a chain, so it had better have an orientation.

But it seems standard to say a simplicial complex is a set equipped with an arbitrary collection of finite subsets. That’s quite pretty, and now the vertices in any simplex (i.e., the points in one of the finite subsets) are unordered. This is the sort of simplicial complex I look at in my paper with Alex Hoffnung.

What do you call an unordered simplicial complex?

How about an “unordered simplicial complex”?

I don’t know all the standard terminology here. But an “unordered simplicial set” is called a symmetric set: it’s a contravariant functor from the category of finite sets to $\mathrm{Set}$. So, if a simplicial complex is a simplicial set with a special property, the analogous thing for symmetric sets deserves to be called a symmetric complex.

I don’t see why orderings are needed for what we’re talking about here: networked dynamical systems. Is that part of what you’re saying?

• Eugene says:

Yes, this is what I have in mind. I have no idea how it relates to electric circuits. I had a pretty decent physics education in high school, which let me coast for the first couple of years of college. But it was so long ago, I don’t remember any E&M any more.

The first thing Lee and I got stuck on is relating maps of simplicial complexes to maps between Hamiltonian systems.
Perhaps we should have tried Lagrangian correspondences, but we didn’t.

The second thing we got stuck on was port-Hamiltonian systems.

• Tobias Fritz says:

Interesting stuff! So for general dynamical systems, you consider directed graphs with the subsystems as vertices, and a arrow from one subsystem to the other whenever the dynamics of the latter depends on the state of the former. For Hamiltonian systems, these relations necessarily always go both ways, and so it’s more natural to consider the simplicial complex of interactions instead. Right?

By the way, concerning the category of simplicial complexes: I’m currently studying The joy of cats (for unrelated reasons). They have some further results on $\mathbf{Simp}$, the category of simplicial complexes:

28.21 Definition: A well-fibred topological construct $(\mathbf{A},U)$ for which $\mathbf{A}$ is a quasitopos is called a topological universe.

[..]

28.23 Examples: $\mathbf{Simp}$ [and some other cats] are topological universes.

7. Eugene says:

John said:

I don’t see why orderings are needed for what we’re talking about here: networked dynamical systems.

What I am trying to say that in networks of Hamiltonian systems you would want to assign potentials to unordered collections of vertices. That means, as you said, that a network in this case is either a decorated symmetric complexes or a decorated symmetric set.

Now, here is an “obvious” interesting question: if these are the objects, what are the morphisms?

If your dynamical system is not Hamiltonian then you could have one part of the system drive another, and that looks directional to me. That’s why Golubitsky and Stewart and their collaborators have directed graphs… (OK, perhaps I should not put it quite so strongly, but it should be roughly correct)

8. We compose morphisms in by composing isomorphism classes of cospans. […]

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