## Structured vs Decorated Cospans

30 January, 2021

Some of us just finished a paper clarifying the connection between two approaches to describing open systems—that is, systems that can interact with their environment, and can be composed to form larger open systems:

• John Baez, Kenny Courser and Christina Vasilakopolou, Structured versus decorated cospans.

And, next week I’m giving a talk about it at YAMCaTS! This is not a conference for felines who like sweet potatoes: it’s the Yorkshire and Midlands Category Seminar, organized by Simona Paoli, Nicola Gambino and Steve Vickers.

In my talk, I’ll start by sketching some ideas behind Halter and Patterson’s software for quickly assembling larger models of COVID-19 from smaller models. Then, I’ll dig deeper into the underlying math, where we use ‘structured’ or ‘decorated’ cospans to model open systems.

This quickly gets into some serious category theory—and since YAMCaTS is a category theory seminar, I won’t shy away from that. Here are my slides:

• John Baez, Structured vs decorated cospans, YAMCaTS, Friday 5 February 2021, 17:00 UTC. Zoom link here, meeting ID 810 4239 7132; to get in use 68302x where x is a one-digit perfect number.

Abstract. One goal of applied category theory is to understand open systems: that is, systems that can interact with the external world. We compare two approaches to describing open systems as cospans equipped with extra data: structured and decorated cospans. Each approach provides a symmetric monoidal double category, and we prove that under certain conditions these symmetric monoidal double categories are isomorphic. We illustrate these ideas with applications to dynamical systems and epidemiological modeling. This is joint work with Kenny Courser and Christina Vasilakopoulou.

I don’t know if my talk will be recorded, but it will be on Zoom so recording it would be easy, and I’ll try to get the organizers to do that.

For videos and slides of two related talks go here:

• Evan Patterson and Micah Halter, Compositional epidemiological modeling using structured cospans.

• John Baez and Kenny Courser, Structured cospans.

• Kenny Courser, Open Systems: a Double Categorical Perspective. (Blog articles here.)

• Michael Shulman, Framed bicategories and monoidal fibrations.

• Joe Moeller and Christina Vasilakopolou, Monoidal Grothendieck construction.

## Open Systems: A Double Categorical Perspective (Part 3)

23 January, 2021

Back to Kenny Courser’s thesis:

• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020.

Last time I explained the problems with decorated cospans as a framework for dealing with open systems. I vaguely hinted that Kenny’s thesis presents two solutions to these problems: so-called ‘structured cospans’, and a new improved approach to decorated cospans. Now let me explain these!

You may wonder why I’m returning to this now, after three months of silence. The reason is that Kenny, Christina Vasilakopolou, and I just finished a paper that continues this story:

• John Baez, Kenny Courser and Christina Vasilakopoulou, Structured versus decorated cospans.

We showed that under certain conditions, structured and decorated cospans are equivalent. So, I’m excited about this stuff again.

Last time I explained Fong’s theorem about decorated cospans:

Fong’s Theorem. Suppose $\mathsf{A}$ is a category with finite colimits, and make $\mathsf{A}$ into a symmetric monoidal category with its coproduct as the tensor product. Suppose $F\colon (\mathsf{A},+) \to (\mathsf{Set},\times)$ is a symmetric lax monoidal functor. Define an F-decorated cospan to be a cospan

in $\mathsf{A}$ together with an element $x\in F(N)$ called a decoration. Then there is a symmetric monoidal category with

• objects of $\mathsf{A}$ as objects,
• isomorphism classes of $F$-decorated cospans as morphisms.

The theorem is true, but it doesn’t apply to all the examples we wanted it to. The problem is that it’s ‘not categorified enough’. It’s fine if we want to decorate the apex $N$ of our cospan with some extra structure: we do this by choosing an element of some set $F(N).$ But in practice, we often want to decorate $N$ with some extra stuff, which means choosing an object of a category $F(N).$ So we should really use not a functor

$F\colon (\mathsf{A},+) \to (\mathsf{Set},\times)$

but something like a functor

$F\colon (\mathsf{A},+) \to (\mathbf{Cat},\times)$

What do I mean by ‘something like a functor?’ Well, $\mathbf{Cat}$ is not just a category but a 2-category: it has categories as objects, functors as morphisms, but also natural transformations as 2-morphisms. The natural notion of ‘something like a functor’ from a category to a 2-category is called a pseudofunctor. And just as we can define symmetric lax monoidal functor, we can define a symmetric lax monoidal pseudofunctor.

All these nuances really matter when we’re studying open graphs, as we were last time!

Here we want the feet of our structured cospan to be finite sets and the apex to be a finite graph. So, we have $\mathsf{A} = \mathsf{FinSet}$ and for any $N \in \mathsf{FinSet}$ we want $F(N)$ to be the set, or category, of finite graphs having $N$ as their set of nodes.

I explained last time all the disasters that ensue when you try to let $F(N)$ be the set of finite graphs having $N$ as its set of nodes. You can try, but you will pay dearly for it! You can struggle and fight, like Hercules trying to chop all the heads off the Hydra, but you still can’t get a symmetric lax monoidal functor

$F\colon (\mathsf{A},+) \to (\mathsf{Set},\times)$

that sends any finite set $N$ to the set of graphs having $N$ as their set of nodes.

But there is a perfectly nice category $F(N)$ of all finite graphs having $N$ as their set of nodes. And you can get a symmetric lax monoidal pseudofunctor

$F\colon (\mathsf{A},+) \to (\mathbf{Cat},\times)$

that sends any any finite set to the category of finite graphs having it as nodes. So you should stop fighting and go with the flow.

Kenny, Christina and I proved an enhanced version of Fong’s theorem that works starting from this more general kind of $F.$ And instead of just giving you a symmetric monoidal category, this theorem gives you a symmetric monoidal double category.

In fact, that is something you should have wanted already, even with Fong’s original hypotheses! The clue is that Fong’s theorem uses isomorphism classes of decorated cospans, which suggests we’d get something better if we used decorated cospans themselves. Kenny tackled this a while ago, getting a version of Fong’s theorem that produces a symmetric monoidal double category, and another version that produces a symmetric monoidal bicategory:

• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 995–1027.

Over the years we’ve realized that the double category is better, because it contains more information and is easier to work with. So, in our new improved approach to decorated cospans, we go straight for the jugular and get a double category. And here’s how it works:

Theorem. Suppose $\mathsf{A}$ is a category with finite colimits, and make $\mathsf{A}$ into a symmetric monoidal category with its coproduct as the tensor product. Suppose $F\colon (\mathsf{A},+) \to (\mathbf{Cat},\times)$ is a symmetric lax monoidal pseudofunctor. Then there is a symmetric monoidal double category $F\mathbb{C}\mathbf{sp}$ in which

• an object is an object of $\mathsf{A}$
• a vertical morphism is a morphism in $\mathsf{A}$
• a horizontal morphism is an F-decorated cospan, meaning a cospan in $\mathsf{A}$ together with a decoration:

• a 2-morphism is a map of decorated cospans, meaning a commutative diagram in $\mathsf{A}$:

together with a morphism $\tau \colon F(h)(x) \to x',$ the map of decorations.

We call $F\mathbb{C}\mathbf{sp}$ a decorated cospan double category. And as our paper explains, this idea lets us fix all the broken attempted applications of Fong’s original decorated cospan categories!

All this is just what any category theorist worth their salt would try, in order to fix the original problems with decorated cospans. It turns out that proving the theorem above is not so easy, mainly because the definition of ‘symmetric monoidal double category’ is rather complex. But if you accept the theorem—including the details of how you get the symmetric monoidal structure on the double category, which I have spared you here—then it doesn’t really matter much that the proof takes work.

Next time I’ll tell you about the other way to fix the original decorated cospan formalism: structured cospans. When these work, they are often easier to use.

Part 1: an overview of Courser’s thesis and related papers.

Part 2: problems with the original decorated cospans.

Part 3: the new improved decorated cospans.

## Categories of Nets (Part 2)

20 January, 2021

guest post by Michael Shulman

Now that John gave an overview of the Petri nets paper that he and I have just written with Jade and Fabrizio, I want to dive a bit more into what we accomplish. The genesis of this paper was a paper written by Fabrizio and several other folks entitled Computational Petri Nets: Adjunctions Considered Harmful, which of course sounds to a category theorist like a challenge. Our paper, and particularly the notion of Σ-net and the adjunction in the middle column relating Σ-nets to symmetric strict monoidal categories, is an answer to that challenge.

Suppose you wanted to “freely” generate a symmetric monoidal category from some combinatorial data. What could that data be? In other words (for a category theorist at least), what sort of category $\mathsf{C}$ appears in an adjunction $\mathsf{C} \rightleftarrows \mathsf{SMC}$? (By the way, all monoidal categories in this post will be strict, so I’m going to drop that adjective for conciseness.)

Perhaps the simplest choice is the same data that naturally generates a plain category, namely a directed graph. However, this is pretty limited in terms of what symmetric monoidal categories it can generate, since the generating morphisms will always only have single generating objects as their domain and codomain.

Another natural choice is the same data that naturally generates a multicategory, which might be called a “multigraph”: a set of objects together with, for every tuple of objects $x_1,\dots,x_n$ and single object $y$, a set of arrows from $(x_1,\dots,x_n)$ to $y$. In the generated symmetric monoidal category, such an arrow gives rise to a morphism $x_1\otimes\cdots\otimes x_n \to y$; thus we can now have multiple generating objects in the domains of generating morphisms, but not the codomains.

Of course, this suggests an even better solution: a set of objects, together with a set of arrows for every pair of tuples $(x_1,\dots,x_m)$ and $(y_1,\dots,y_n)$. I’d be tempted to call this a “polygraph”, since it also naturally generates a polycategory. But other folks got there first and called it a “tensor scheme” and also a “pre-net”. In the latter case, the objects are called “places” and the morphisms “transitions”. But whatever we call it, it allows us to generate free symmetric monoidal categories in which the domains and codomains of generating morphisms can both be arbitrary tensor products of generating objects. For those who like fancy higher-categorical machinery, it’s the notion of computad obtained from the monad for symmetric monoidal categories.

However, pre-nets are not without flaws. One of the most glaring, for people who actually want to compute with freely generated symmetric monoidal categories, is that there aren’t enough morphisms between them. For instance, suppose one pre-net $N$ has three places $x,y,z$ and a transition $f:(x,x,y) \to z$, while a second pre-net $N'$ has three places $x',y',z'$ and a transition $f':(x',y',x') \to z'$. Once we generate a symmetric monoidal category, then $f$ can be composed with a symmetry $x\otimes y \otimes x \cong x\otimes x\otimes y$ and similarly for $f'$; so the symmetric monoidal categories generated by $N$ and $N'$ are isomorphic. But there isn’t even a single map of pre-nets from $N$ to $N'$ or vice versa, because a map of pre-nets has to preserve the ordering on the inputs and outputs. This is weird and annoying for combinatorial data that’s supposed to present a symmetric monoidal category.

Another way of making essentially the same point is that just as the adjunction between SMCs and directed graphs factors through categories, and the adjunction between SMCs and multigraphs factors through multicategories, the adjunction between SMCs and pre-nets factors through non-symmetric monoidal categories. In other words, a pre-net is really better viewed as data for generating a non-symmetric monoidal category, which we can then freely add symmetries to.

By contrast, in the objects that we call “Petri nets”, the domain and codomain of each generating morphism are elements of the free commutative monoid on the set of places—as opposed to elements of the free monoid, which is what they are for a pre-net. Thus, the domain of $f$ and $f'$ above would be $x+x+y$ and $x+y+x$ respectively, which in a commutative monoid are equal (both are $2x+y$). So the corresponding Petri nets of $N$ and $N'$ are indeed isomorphic. However, once we squash everything down in this way, we lose the ability to functorially generate a symmetric monoidal category; all we can generate is a commutative monoidal category where all the symmetries are identities.

At this point we’ve described the upper row and the left- and right-hand columns in John’s diagram:

What’s missing is a kind of net in the middle that corresponds to symmetric monoidal categories. To motivate the definition of Σ-net, consider how to solve the problem above of the “missing morphisms”. We want to send $f:(x,x,y) \to z$ to a “permuted version” of $f':(x',y',x') \to z'$. For this to be implemented by an actual set-map, we need this “permuted version” to be present in the data of $N'$ somehow. This suggests that the transitions should come with a permutation action like that of, say, a symmetric multicategory. Then inside $N'$ we can actually act on $f'$ by the transposition $\tau = (2,3) \in S_3$, yielding a new morphism $\tau(f') : (x',x',y')\to z'$ which we can take to be the image of $f$. Of course, we can also act on $f'$ by other permutations, and likewise on $f$; but since these permutation actions are part of the structure they must be preserved by the morphism, so sending $f$ to $\tau(f')$ uniquely determines where we have to send all these permutation images.

Now you can go back and look again at John’s definition of Σ-net: a set $S$, a groupoid $T$, and a discrete opfibration $T \to P S \times P S ^{op}$, where $P$ denotes the free-symmetric-strict-monoidal-category functor $\mathsf{Set} \to \mathsf{Cat}$. Such a discrete opfibration is the same as a functor $N \colon P S \times P S ^{op} \to \mathsf{Set}$, and the objects of $P S$ are the finite sequences of elements of $S$ while its morphisms are permutations; thus this is precisely a pre-net (the action of the functor $N$ on objects) with permutation actions as described above. I won’t get into the details of constructing the adjunction relating Σ-nets to symmetric monoidal categories; you can read the paper, or maybe I’ll blog about it later.

However, in solving the “missing morphisms” problem, we’ve introduced a new possibility. Suppose we act on $f \colon (x,x,y) \to z$ by the transposition $\sigma = (1,2) \in S_3$ that switches the first two entries. We get another transition $(x,x,y)\to z$ with the same domain and codomain as $f$; so it might equal $f$, or it might not! In other words, transitions in a Σ-net can have isotropy. If $\sigma(f)=f$, then when we generate a free symmetric monoidal category from our Σ-net, the corresponding morphism $f:x\otimes x \otimes y \to z$ will have the property that when we compose it with the symmetry morphism $x\otimes x\otimes y \cong x\otimes x\otimes y$ we get back $f$ again. No symmetric monoidal category generated by a pre-net has this property; it’s more like the behavior of the commutative monoidal category generated by a Petri net, except that in the latter case the symmetry $x\otimes x\otimes y \cong x\otimes x\otimes y$ itself is the identity, rather than just acting by the identity on $f$.

This suggests that Σ-nets can either “behave like pre-nets” or “behave like Petri nets”. This is made precise by the bottom row of adjunctions in the diagram. On one hand, we can map a pre-net to a Σ-net by freely generating the action of all permutations. This has a right adjoint that just forgets the permutation action (which actually has a further right adjoint, although that’s a bit weird). On the other hand, we can map a Petri net to a Σ-net by making all the permutations act as trivially as possible; this has a left adjoint that identifies each transition with all its permutation images. And these adjunctions commute with the three “free monoidal category” adjunctions in reasonable ways (see the paper for details).

The right adjoint mapping Petri nets into Σ-nets is fully faithful, so we really can say that Σ-nets “include” Petri nets. The left adjoint mapping pre-nets to Σ-nets is not fully faithful—it can’t possibly be, since the whole point of introducing Σ-nets was that pre-nets don’t have enough morphisms! But the full image of this functor is equivalent to a fourth kind of net: Kock’s whole-grain Petri nets. Kock’s approach to solving the problem of pre-nets is somewhat different, more analogous to the notion of “fat” symmetric monoidal category: he takes the domain and codomain of each transition to be a family of places indexed by a finite set. But his category turns out to be equivalent to the category of Σ-nets that are freely generated by some pre-net. (Kock actually proved this himself, as well as sketching the adjunction between Σ-nets and symmetric monoidal categories. He called Σ-nets “digraphical species”.)

So Σ-nets “include” both Petri nets and pre-nets, in an appropriate sense. The pre-nets (or, more precisely, whole-grain nets) are the Σ-nets with free permutation actions (trivial isotropy), while the Petri nets are the Σ-nets with trivial permutation actions (maximal isotropy). In Petri-net-ese, these correspond to the “individual token philosophy” and the “collective token philosophy”, respectively. (This makes it tempting to refer to the functors from Σ-nets to pre-nets and Petri nets as individuation and collectivization respectively.) But Σ-nets also allow us to mix and match the two philosophies, having some transitions with trivial isotropy, others with maximal isotropy, and still others with intermediate isotropy.

I like to think of Σ-nets as a Petri net analogue of orbifolds. Commutative-monoid-based Petri nets are like “coarse moduli spaces”, where we’ve quotiented by all symmetries but destroyed all the isotropy information; while whole-grain Petri nets are like manifolds, where we have no singularities but can only quotient by free actions. Pre-nets can then be thought of a “presentation” of a manifold, such as by a particular way of gluing coordinate patches together: useful in concrete examples, but not the “invariant” object we really want to study mathematically.

Part 1: three kinds of nets, and the kinds of monoidal categories they generate.

Part 2: what kind of net is best for generating symmetric monoidal categories?

## Epidemiological Modeling With Structured Cospans

19 October, 2020

This is a wonderful development! Micah Halter and Evan Patterson have taken my work on structured cospans with Kenny Courser and open Petri nets with Jade Master, together with Joachim Kock’s whole-grain Petri nets, and turned them into a practical software tool!

Then they used that to build a tool for ‘compositional’ modeling of the spread of infectious disease. By ‘compositional’, I mean that they make it easy to build more complex models by sticking together smaller, simpler models.

Even better, they’ve illustrated the use of this tool by rebuilding part of the model that the UK has been using to make policy decisions about COVID19.

All this software was written in the programming language Julia.

I had expected structured cospans to be useful in programming and modeling, but I didn’t expect it to happen so fast!

For details, read this great article:

• Micah Halter and Evan Patterson, Compositional epidemiological modeling using structured cospans, 17 October 2020.

Abstract. The field of applied category theory (ACT) aims to put the compositionality inherent to scientific and engineering processes on a firm mathematical footing. In this post, we show how the mathematics of ACT can be operationalized to build complex epidemiological models in a compositional way. In the first two sections, we review the idea of structured cospans, a formalism for turning closed systems into open ones, and we illustrate its use in Catlab through the simple example of open graphs. Finally, we put this machinery to work in the setting of Petri nets and epidemiological models. We construct a portion of the COEXIST model for the COVID-19 pandemic and we simulate the resulting ODEs.

You can see related articles by James Fairbanks, Owen Lynch and Evan Patterson here:

Also try these videos:

• James Fairbanks, AlgebraicJulia: Applied category theory in Julia, 29 July 2020.

• Evan Patterson, Realizing applied category theory in Julia, 16 January 2020.

I’m biased, but I think this is really cool cutting-edge stuff. If you want to do work along these lines let me know here and I’ll get Patterson to take a look.

Here’s part of a network created using their software:

## Open Petri Nets and Their Categories of Processes

14 October, 2020

My student Jade Master will be talking about her work on open Petri nets at the online category theory seminar at UNAM on Wednesday October 21st at 18:00 UTC (11 am Pacific Time):

Open Petri Nets and Their Categories of Processes

Abstract. In this talk we will discuss Petri nets from a categorical perspective. A Petri net freely generates a symmetric monoidal category whose morphisms represent its executions. We will discuss how to make Petri nets ‘open’—i.e., equip them with input and output boundaries where resources can flow in and out. Open Petri nets freely generate open symmetric monoidal categories: symmetric monoidal categories which can be glued together along a shared boundary. The mapping from open Petri nets to their open symmetric monoidal categories is functorial and this gives a compositional framework for reasoning about the executions of Petri nets.

You can see the talk live, or later recorded, here:

• John Baez and Jade Master, Open Petri nets.

• Jade Master, Generalized Petri nets.

You can see Jade’s slides for a related talk here:

Abstract. The reachability semantics for Petri nets can be studied using open Petri nets. For us an ‘open’ Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri}),$ which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\mathbf{pen}(\mathsf{Petri}).$ Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of $\mathbb{O}\mathbf{pen}(\mathsf{Petri}).$ Here we describe the reachability semantics, which assigns to each open Petri net the relation saying which markings of the outputs can be obtained from a given marking of the inputs via a sequence of transitions. We show this semantics gives a symmetric monoidal lax double functor from $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$ to the double category of relations. A key step in the proof is to treat Petri nets as presentations of symmetric monoidal categories; for this we use the work of Meseguer, Montanari, Sassone and others.

## Network Models

7 October, 2020

Good news: my student Joe Moeller will be taking a job at NIST, the National Institute of Standards and Technology! He’ll be working with Spencer Breiner and Eswaran Subrahmanian on categories in engineering and system design.

Joe Moeller will be talking about his work on ‘network models’ at the online category theory seminar at UNAM on Wednesday October 14th at 18:00 UTC (11 am Pacific Time):

Network Models

Abstract. Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce `network models’ to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to Cat, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.

You can watch the talk here:

and here’s the original paper:

• John Baez, John Foley, Blake Pollard and Joseph Moeller, Network models, Theory and Applications of Categories 35 (2020), 700–744.

## Open Systems: A Double Categorical Perspective (Part 2)

16 September, 2020

Back to Kenny Courser’s thesis:

• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020.

One thing Kenny does here is explain the flaws in a well-known framework for studying open systems: decorated cospans. Decorated cospans were developed by my student Brendan Fong. Since I was Brendan’s advisor at the time, a hefty helping of blame for not noticing the problems belongs to me! But luckily, Kenny doesn’t just point out the problems: he shows how to fix them. As a result, everything we’ve done with decorated cospans can be saved.

The main theorem on decorated cospans is correct; it’s just less useful than we’d hoped! The idea is to cook up a category where the morphisms are open systems. The objects of this category could be something simple like sets, but morphisms from $X$ to $Y$ could be something more interesting, like ‘open graphs’:

Here $X$ and $Y$ are mapped into a third set in the middle, but this set in the middle is the set of nodes of a graph. We say the set in the middle has been ‘decorated’ with the structure of a graph.

Here’s how the original theory of decorated cospans seeks to make this precise.

Fong’s Theorem. Suppose $\mathsf{A}$ is a category with finite colimits, and make $\mathsf{A}$ into a symmetric monoidal category with its coproduct as the tensor product. Suppose $F\colon (\mathsf{A},+) \to (\mathsf{Set},\times)$ is a symmetric lax monoidal functor. Define an F-decorated cospan to be a cospan

in $\mathsf{A}$ together with an element of $F(N).$ Then there is a symmetric monoidal category with

• objects of $\mathsf{A}$ as objects,
• isomorphism classes of $F$-decorated cospans as morphisms.

I won’t go into many details, but let me say how to compose two decorated spans, and also how this ‘isomorphism class’ business works.

Given two decorated cospans we compose their underlying cospans in the usual way, via pushout:

We get a cospan from $X$ to $Z.$ To decorate this we need an element of $F(M +_Y N).$ So, we take the decorations we have on the cospans being composed, which together give an element of $F(N) \times F(M),$ and apply this composite map:

$F(N) \times F(M) \longrightarrow F(N+M) \longrightarrow F(N+_Y M)$

Here the first map, called the laxator, comes from the fact that $F$ is a lax monoidal functor, while the second comes from applying $F$ to the canonical map $N+M \to N+_Y M.$

Since composing cospans involves a pushout, which is defined via a universal property, the composite is only well-defined up to isomorphism. So, to get an actual category, we take isomorphism classes of decorated cospans as our morphisms.

Here an isomorphism of cospans is a commuting diagram like this:

where $h$ is an isomorphism. If the first cospan here has a decoration $d \in F(N)$ and the second has a decoration $d' \in F(N'),$ then we have an isomorphism of decorated cospans if $F(h)(d) = d'.$

So, that’s the idea. The theorem is true, and it works fine for some applications—but not so well for others, like the example of open graphs!

Why not? Well, let’s look at this example in more detail. Given a finite set $N,$ let’s define a graph on $N$ to be a finite set $E$ together with two functions $s, t \colon E \to N.$ We call $N$ the set of nodes, $E$ the set of edges, and the functions $s$ and $t$ map each edge to its source and target, respectively. So, a graph on $N$ is a way of choosing a graph whose set of nodes is $N.$

We can try to apply the above theorem taking

$\mathsf{A} = \mathsf{FinSet}$

and letting

$F \colon \mathsf{FinSet} \to \mathsf{Set}$

be the functor sending each finite set $N$ to the set of all graphs on $N.$

The first problem, which Brendan and I noticed right away, is that there’s not really a set of graphs on $N.$ There’s a proper class! $E$ ranges over all possible finite sets, and there’s not a set of all finite sets.

This should have set alarm bells ringing right away. But we used a standard dodge. In fact there are two. One is to replace $\mathsf{FinSet}$ with an equivalent small category, and define a graph just as before but taking $N$ and $E$ to be objects in this equivalent category. Another is to invoke the axiom of universes. Either way, we get a set of graphs on each $N.$

Then Fong’s theorem applies, and we get a decorated cospan category with:

• ‘finite sets’ as objects,
• isomorphism classes of open graphs as morphisms.

Here I’m putting ‘finite sets’ in quotes because of the trickery I just mentioned, but it’s really not a big deal so I’ll stop now. An open graph has a finite set $N$ of nodes, a finite set $E$ of edges, maps $s,t \colon E \to N$ saying the source and target of each edge, and two maps $f \colon X \to N$ and $g \colon Y \to N.$

These last two maps are what make it an open graph going from $X$ to $Y:$

Isomorphism classes of open graphs from $X$ to $Y$ are the morphisms from $X$ to $Y$ in our decorated cospan category.

But later, Kenny and I noticed a second problem. The concept of ‘isomorphic decorated cospan’ doesn’t behave well in this example: the concept of isomorphism is too narrowly defined!

Suppose you and I have isomorphic decorated cospans:

In the example at hand, this means you have a graph on the finite set $N$ and I have a graph on the finite set $N'.$ Call yours $d \in F(N)$ and mine $d' \in F(N').$

We also have a bijection $h \colon N \to N'$ such that

$F(h)(d) = d'$

What does this mean? I haven’t specified the functor $F$ in detail so you’ll have to take my word for it, but it should be believable. It means that my graph is exactly like yours except that we replace the nodes of your graph, which are elements of $N,$ by the elements of $N'$ that they correspond to. But this means the edges of my graph must be exactly the same as the edges of yours. It’s not good enough for our graphs to have isomorphic sets of edges: they need to be equal!.

For a more precise account of this, with pictures, read the introduction to Chapter 3 of Kenny’s thesis.

So, our decorated cospan category has ‘too many morphisms’. Two open graphs will necessarily define different morphisms if they have different sets of edges.

This set Kenny and I to work on a new formalism, structured cospans, that avoids this problem. Later Kenny and Christina Vasilakopoulou also figured out a way to fix the decorated cospan formalism. Kenny’s thesis explains all this, and also how structured cospans are related to the ‘new, improved’ decorated cospans.

But then something else happened! Christina Vasilakopoulou was a postdoc at U.C. Riverside while all this was going on. She and my grad student Joe Moeller wrote a paper on something called the monoidal Grothendieck construction, which plays a key role in relating structured cospans to the new decorated cospans. But the anonymous referee of their paper pointed out another problem with the old decorated cospans!

Briefly, the problem is that the functor

$F \colon (\mathsf{FinSet},+) \to (\mathsf{Set},\times)$

that sends each $N$ to the set of graphs having $N$ as their set of vertices cannot be made lax monoidal in the desired way. To make $F$ lax monoidal, we must pick a natural transformation called the laxator:

$\phi_{N,M} \colon F(N) \times F(M) \to F(N+M)$

I used this map earlier when explaining how to compose decorated cospans.

The idea seems straightforward enough: given a graph on $N$ and a graph on $M$ we get a graph on their disjoint union $N+M.$ This is true, and there is a natural transformation $\phi_{N,M}$ that does this.

But the definition of lax monoidal functor demands that the laxator make a certain hexagon commute! And it does not!

I won’t draw this hexagon here; you can see it at the link or in the intro to Chapter 3 of Kenny’s thesis, where he explains this problem. The problem arises because when we have three sets of edges, say $E,E',E'',$ we typically have

$(E + E') + E'' \ne E + (E' + E'')$

There is a sneaky way to get around this problem, which he also explains: define graphs using a category equivalent to $\mathsf{FinSet}$ where $+$ is strictly associative, not just up to isomorphism!

This is good enough to make $F$ lax monoidal. But Kenny noticed yet another problem: $F$ is still not symmetric lax monoidal. If you try to show it is, you wind up needing two graphs to be equal: one with $E+E'$ as its set of edges, and another with $E'+E$ as its set of edges. These aren’t equal! And at this point it seems we hit a dead end. While there’s a category equivalent to $\mathsf{FinSet}$ where $+$ is strictly associative, there’s no such category where $+$ is also strictly commutative.

In the end, by going through all these contortions, we can use a watered-down version of Fong’s theorem to get a category with open graphs as morphisms, and we can make it lax monoidal—but not symmetric lax monoidal.

It’s clear that something bad is going on here. We are not following the tao of mathematics. We keep wanting things to be equal, that should only be isomorphic. The problem is that we’re treating a graph on a set as extra structure on that set, when it’s actually extra stuff.

Structured cospans, and the new decorated cospans, are the solution! For example, the new decorated cospans let us use a category of graphs with a given set of nodes, instead of a mere set. Now $F(N)$ is a category rather than a set. This category lets us talk about isomorphic graphs with the same set of vertices, and all our problems evaporate.

Part 1: an overview of Courser’s thesis and related papers.

Part 2: problems with the original decorated cospans.

Part 3: the new improved decorated cospans.

## Open Systems: A Double Categorical Perspective (Part 1)

15 August, 2020

My student Kenny Courser’s thesis has hit the arXiv:

• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020.

He’s been the driving force behind a lot of work on open systems and networks at U. C. Riverside. By the way, he’s looking for a job, so if you think you know a position that’s good for someone who can teach all kinds of math and also strong on applied category theory, give him or me a shout.

But let me describe his thesis.

His thesis is big! It lays out a general approach to open systems—systems that can interact with their environment. In this approach, you can attach open systems in series to form larger open systems, so they act like morphisms in a category:

But you can also study 2-morphisms between open systems. These describe ways to include a little system in a bigger one, or simplify a big complicated system down to smaller one:

To handle all this, Courser uses ‘double categories’, which he explains.

His formalism also lets you set two open systems side by side ‘in parallel’ and get a new open system:

To handle this, he uses ‘symmetric monoidal double categories’. He explains what these are, and how to get them. And he illustrates his setup with examples:

At a more technical level, Courser explains the problems with Brendan Fong’s and my work on decorated cospans and shows how to fix in not just one but two ways: using structured cospans, and using a new improved version of decorated cospans. He also shows that these two approaches are equivalent under fairly general conditions.

His thesis unifies a number of papers:

• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 995–1027.

• John Baez and Kenny Courser, Coarse-graining open Markov processes, Theory and Applications of Categories 33 (2018), 1223–1268. (Blog article here.)

• John Baez and Kenny Courser, Structured cospans, Theory and Applications of Categories 35 (2020), 1771–1822. (Blog article here.)

• John Baez, Kenny Courser and Christina Vasilakopoulou, Structured versus decorated cospans.

The last introduces the new improved decorated cospans and proves their equivalence to structured cospans under some conditions.

Next time I’ll explain the problems with the original decorated cospan formalism. Another nice thing about Kenny’s thesis is that it goes over a bunch of papers that were afflicted by these problems, and shows how to fix them.

Part 1: an overview of Courser’s thesis and related papers.

Part 2: problems with the original decorated cospans.

Part 3: the new improved decorated cospans.

## Linear Logic and Petri Nets

28 July, 2020

Wow! Elena Di Lavore and Xiaoyan Li explained how to make a category of Petri nets that’s a model of linear logic! I consider myself a sort of expert on Petri nets, but I didn’t know this stuff:

• Elena Di Lavore and Xiaoyan Li, Linear logic flavoured composition of Petri nets, The n-Category Café, 27 July 2020.

It has great pictures, too. Let me summarize a tiny bit.

A Petri net is a very simple thing. Here’s a Petri net that shows how healthy white blood cells (H), individual viruses (V) and infected white blood cells (I) interact when someone gets AIDS:

The yellow boxes are different kinds of things, called species; the aqua boxes are processes, called transitions.

There are different ways to form categories using Petri nets. Jade Master and I have focused on two:

1) How to turn a Petri net into a category where the morphisms say what the Petri net can do.

2) How to make a category with ‘open’ Petri nets as morphisms. Composing these lets you build big Petri nets from smaller pieces.

Di Lavore and Li instead explain:

3) How to make a category with elementary Petri nets as objects: a morphism from one elementary Petri net to another turns each transition of the first into a transition of the second.

An elementary Petri net is one where each transition uses each species at most once as an input and at most once as an output:

The category they get has lots of interesting structure! Like products, shown here:

In fact it has products, coproducts, two other monoidal structures, and exponentials—all fitting together in a wonderful way, as described by intuitionistic linear logic! To prove this, the key is to use Valeria de Paiva’s work on “Dialectica categories”. They explain how.

This is not original research: Elena Di Lavore and Xiaoyan Li wrote this blog article for the ACT2020 Adjoint School, and they’re explaining Carolyn Brown and Doug Gurr’s paper “A categorical linear framework for Petri nets”.

It’s worth comparing this paper:

• Uffe Engberg and Glynn Winskel, Petri nets as models of linear logic, in Colloquium on Trees in Algebra and Programming, Springer, Berlin, 1990, pp. 147–161.

Engberg and Winskel get a model of linear logic—or to be precise, a ‘commutative quantale’—by taking the category you get from a single Petri net as in item 1) and massaging it a bit. I explained it here:

• John Baez, Quantales from Petri nets, Azimuth, 6 October 2019.

## A Localic Approach to Dependency, Conflict, and Concurrency

28 April, 2020

In the fifth talk of the ACT@UCR seminar, Gershom Bazerman told how to use locales to study the semantics of dependency, conflict, and concurrency.

Afterwards we discussed his talk at the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/April.2029th.3A.20Gershom.20Bazerman

You can view or join the conversation there if you sign in.

You can see his slides here, or download a video here, or watch the video here:

• Gershom Bazerman, A localic approach to the semantics of dependency, conflict, and concurrency.

Abstract. Petri nets have been of interest to applied category theory for some time. Back in the 1980s, one approach to their semantics was given by algebraic gadgets called “event structures.” We use classical techniques from order theory to study event structures without conflict restrictions (which we term “dependency structures with choice”) by their associated “traces”, which let us establish a one-to-one correspondence between DSCs and a certain class of locales. These locales have an internal logic of reachability, which can be equipped with “versioning” modalities that let us abstract away certain unnecessary detail from an underlying DSC. With this in hand we can give a general notion of what it means to “solve a dependency problem” and combinatorial results bounding the complexity of this. Time permitting, I will sketch work-in-progress which hopes to equip these locales with a notion of conflict, letting us capture the full semantics of general event structures in the form of homological data, thus providing one avenue to the topological semantics of concurrent systems. This is joint work with Raymond Puzio.