## Applied Category Theory at UCR (Part 3)

13 November, 2017

We had a special session on applied category theory here at UCR:

Applied category theory, Fall Western Sectional Meeting of the AMS, 4-5 November 2017, U.C. Riverside.

A bunch of people stayed for a few days afterwards, and we had a lot of great discussions. I wish I could explain everything that happened, but I’m too busy right now. Luckily, even if you couldn’t come here, you can now see slides of almost all the talks… and videos of many!

Click on talk titles to see abstracts. For multi-author talks, the person whose name is in boldface is the one who gave the talk. For videos, go here: I haven’t yet created links to all the videos.

### Saturday November 4, 2017

9:00 a.m.A higher-order temporal logic for dynamical systemstalk slides.
David I. Spivak, MIT.

10:00 a.m.
Algebras of open dynamical systems on the operad of wiring diagramstalk slides.
Dmitry Vagner, Duke University
David I. Spivak, MIT
Eugene Lerman, University of Illinois at Urbana-Champaign

10:30 a.m.
Abstract dynamical systemstalk slides.
Christina Vasilakopoulou, UCR
David Spivak, MIT
Patrick Schultz, MIT

3:00 p.m.
Decorated cospanstalk slides.
Brendan Fong, MIT

4:00 p.m.
Compositional modelling of open reaction networkstalk slides.
Blake S. Pollard, UCR
John C. Baez, UCR

4:30 p.m.
A bicategory of coarse-grained Markov processestalk slides.
Kenny Courser, UCR

5:00 p.m.
A bicategorical syntax for pure state qubit quantum mechanicstalk slides.
Daniel M. Cicala, UCR

5:30 p.m.
Open systems in classical mechanicstalk slides.

### Sunday November 5, 2017

9:00 a.m.
Controllability and observability: diagrams and dualitytalk slides.
Jason Erbele, Victor Valley College

9:30 a.m.
Frobenius monoids, weak bimonoids, and corelationstalk slides.
Brandon Coya, UCR

10:00 a.m.
Compositional design and tasking of networks.
John D. Foley, Metron, Inc.
John C. Baez, UCR
Joseph Moeller, UCR
Blake S. Pollard, UCR

10:30 a.m.
Joseph Moeller, UCR
John Foley, Metron Inc.
John C. Baez, UCR
Blake S. Pollard, UCR

2:00 p.m.
Reeb graph smoothing via cosheavestalk slides.
Vin de Silva, Department of Mathematics, Pomona College

3:00 p.m.
Knowledge representation in bicategories of relationstalk slides.
Evan Patterson, Stanford University, Statistics Department

3:30 p.m.
The multiresolution analysis of flow graphstalk slides.
Steve Huntsman, BAE Systems

4:00 p.m.
Data modeling and integration using the open source tool Algebraic Query Language (AQL)talk slides.
Peter Y. Gates, Categorical Informatics
Ryan Wisnesky, Categorical Informatics

## Complex Adaptive Systems (Part 6)

31 October, 2017

I’ve been slacking off on writing this series of posts… but for a good reason: I’ve been busy writing a paper on the same topic! In the process I caught a couple of mistakes in what I’ve said so far. But more importantly, there’s a version out now, that you can read:

• John Baez, John Foley, Blake Pollard and Joseph Moeller, Network models.

There will be two talks about this at the AMS special session on Applied Category Theory this weekend at U. C. Riverside: one by John Foley of Metron Inc., and one by my grad student Joseph Moeller. I’ll try to get their talk slides someday. But for now, here’s the basic idea.

Our goal is to build operads suited for designing networks. These could be networks where the vertices represent fixed or moving agents and the edges represent communication channels. More generally, they could be networks where the vertices represent entities of various types, and the edges represent relationships between these entities—for example, that one agent is committed to take some action involving the other. This paper arose from an example where the vertices represent planes, boats and drones involved in a search and rescue mission in the Caribbean. However, even for this one example, we wanted a flexible formalism that can handle networks of many kinds, described at a level of detail that the user is free to adjust.

To achieve this flexibility, we introduced a general concept of ‘network model’. Simply put, a network model is a kind of network. Any network model gives an operad whose operations are ways to build larger networks of this kind by gluing smaller ones. This operad has a ‘canonical’ algebra where the operations act to assemble networks of the given kind. But it also has other algebras, where it acts to assemble networks of this kind equipped with extra structure and properties. This flexibility is important in applications.

What exactly is a ‘kind of network’? That’s the question we had to answer. We started with some examples, At the crudest level, we can model networks as simple graphs. If the vertices are agents of some sort and the edges represent communication channels, this means we allow at most one channel between any pair of agents.

However, simple graphs are too restrictive for many applications. If we allow multiple communication channels between a pair of agents, we should replace simple graphs with ‘multigraphs’. Alternatively, we may wish to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. This requires replacing simple graphs with ‘directed graphs’. To combine these features we could use ‘directed multigraphs’.

But none of these are sufficiently general. It’s also important to consider graphs with colored vertices, to specify different types of agents, and colored edges, to specify different types of channels. This leads us to ‘colored directed multigraphs’.

All these are examples of what we mean by a ‘kind of network’, but none is sufficiently general. More complicated kinds, such as hypergraphs or Petri nets, are likely to become important as we proceed.

Thus, instead of separately studying all these kinds of networks, we introduced a unified notion that subsumes all these variants: a ‘network model’. Namely, given a set $C$ of ‘vertex colors’, a network model is a lax symmetric monoidal functor

$F: \mathbf{S}(C) \to \mathbf{Cat}$

where $\mathbf{S}(C)$ is the free strict symmetric monoidal category on $C$ and $\mathbf{Cat}$ is the category of small categories.

Unpacking this somewhat terrifying definition takes a little work. It simplifies in the special case where $F$ takes values in $\mathbf{Mon},$ the category of monoids. It simplifies further when $C$ is a singleton, since then $\mathbf{S}(C)$ is the groupoid $\mathbf{S},$ where objects are natural numbers and morphisms from $m$ to $n$ are bijections

$\sigma: \{1,\dots,m\} \to \{1,\dots,n\}$

If we impose both these simplifying assumptions, we have what we call a one-colored network model: a lax symmetric monoidal functor

$F : \mathbf{S} \to \mathbf{Mon}$

As we shall see, the network model of simple graphs is a one-colored network model, and so are many other motivating examples. If you like André Joyal’s theory of ‘species’, then one-colored network models should be pretty fun, since they’re species with some extra bells and whistles.

But if you don’t, there’s still no reason to panic. In relatively down-to-earth terms, a one-colored network model amounts to roughly this. If we call elements of $F(n)$ ‘networks with $n$ vertices’, then:

• Since $F(n)$ is a monoid, we can overlay two networks with the same number of vertices and get a new one. We call this operation

$\cup \colon F(n) \times F(n) \to F(n)$

• Since $F$ is a functor, the symmetric group $S_n$ acts on the monoid $F(n).$ Thus, for each $\sigma \in S_n$, we have a monoid automorphism that we call simply

$\sigma \colon F(n) \to F(n)$

• Since $F$ is lax monoidal, we also have an operation

$\sqcup \colon F(m) \times F(n) \to F(m+n)$

We call this operation the disjoint union of networks. In examples like simple graphs, it looks just like what it sounds like.

Unpacking the abstract definition further, we see that these operations obey some equations, which we list in Theorem 11 of our paper. They’re all obvious if you draw pictures of examples… and don’t worry, our paper has a few pictures. (We plan to add more.) For example, the ‘interchange law’

$(g \cup g') \sqcup (h \cup h') = (g \sqcup h) \cup (g' \sqcup h')$

holds whenever $g,g' \in F(m)$ and $h, h' \in F(n).$ This is a nice relationship between overlaying networks and taking their disjoint union.

In Section 2 of our apper we study one-colored network models, and give lots of examples. In Section 3 we describe a systematic procedure for getting one-colored network models from monoids. In Section 4 we study general network models and give examples of these. In Section 5 we describe a category $\mathbf{NetMod}$ of network models, and show that the procedure for getting network models from monoids is functorial. We also make $\mathbf{NetMod}$ into a symmetric monoidal category, and give examples of how to build new networks models by tensoring old ones.

Our main result is that any network model gives a typed operad, also known as a ‘colored operad’. This operad has operations that describe how to stick networks of the given kind together to form larger networks of this kind. This operad has a ‘canonical algebra’, where it acts on networks of the given kind—but the real point is that it has lots of other algebra, where it acts on networks of the given kind equipped with extra structure and properties.

The technical heart of our paper is Section 6, mainly written by Joseph Moeller. This provides the machinery to construct operads from network models in a functorial way. Category theorists should find this section interesting, because because it describes enhancements of the well-known ‘Grothendieck construction’ of the category of elements $\int F$ of a functor

$F: \mathbf{C} \to \mathbf{Cat}$

where $\mathbf{C}$ is any small category. For example, if $\mathbf{C}$ is symmetric monoidal and $F : \mathbf{C} \to (\mathbf{Cat}, \times)$ is lax symmetric monoidal, then we show $\int F$ is symmetric monoidal. Moreover, we show that the construction sending the lax symmetric monoidal functor $F$ to the symmetric monoidal category $\int F$ is functorial.

In Section 7 we apply this machinery to build operads from network models. In Section 8 we describe some algebras of these operads, including an algebra whose elements are networks of range-limited communication channels. In future work we plan to give many more detailed examples, and to explain how these algebras, and the homomorphisms between them, can be used to design and optimize networks.

I want to explain all this in more detail—this is a pretty hasty summary, since I’m busy this week. But for now you can read the paper!

## Applied Category Theory 2018 — Adjoint School

22 October, 2017

The deadline for applying to this ‘school’ on applied category theory is Wednesday November 1st.

Applied Category Theory: Adjoint School: online sessions starting in January 2018, followed by a meeting 23–27 April 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The name ‘adjoint school’ is a bad pun, but the school should be great. Here’s how it works:

### Overview

The Workshop on Applied Category Theory 2018 takes place in May 2018. A principal goal of this workshop is to bring early career researchers into the applied category theory community. Towards this goal, we are organising the Adjoint School.

The Adjoint School will run from January to April 2018. By the end of the school, each participant will:

• be familiar with the language, goals, and methods of four prominent, current research directions in applied category theory;
• have worked intensively on one of these research directions, mentored by an expert in the field; and
• know other early career researchers interested in applied category theory.

They will then attend the main workshop, well equipped to take part in discussions across the diversity of applied category theory.

### Structure

The Adjoint School comprises (1) an Online Reading Seminar from January to April 2018, and (2) a four day Research Week held at the Lorentz Center, Leiden, The Netherlands, from Monday April 23rd to Thursday April 26th.

In the Online Reading Seminar we will read papers on current research directions in applied category theory. The seminar will consist of eight iterations of a two week block. Each block will have one paper as assigned reading, two participants as co-leaders, and three phases:

• A presentation (over WebEx) on the assigned reading delivered by the two block co-leaders.
• Reading responses and discussion on a private forum, facilitated by Brendan Fong and Nina Otter.
• Publication of a blog post on the n-Category Café written by the co-leaders.

Each participant will be expected to co-lead one block.

The Adjoint School is taught by mentors John Baez, Aleks Kissinger, Martha Lewis, and Pawel Sobocinski. Each mentor will mentor a working group comprising four participants. During the second half of the Online Reading Seminar, these working groups will begin to meet with their mentor (again over video conference) to learn about open research problems related to their reading.

In late April, the participants and the mentors will convene for a four day Research Week at the Lorentz Center. After opening lectures by the mentors, the Research Week will be devoted to collaboration within the working groups. Morning and evening reporting sessions will keep the whole school informed of the research developments of each group.

The following week, participants will attend Applied Category Theory 2018, a discussion-based 60-attendee workshop at the Lorentz Center. Here they will have the chance to meet senior members across the applied category theory community and learn about ongoing research, as well as industry applications.

Following the school, successful working groups will be invited to contribute to a new, to be launched, CUP book series.

Meetings will be on Mondays; we will determine a time depending on the locations of the chosen participants.

### Research projects

John Baez: Semantics for open Petri nets and reaction networks
Petri nets and reaction networks are widely used to describe systems of interacting entities in computer science, chemistry and other fields, but the study of open Petri nets and reaction networks is new, and raise many new questions connected to Lawvere’s “functorial semantics”.

Aleks Kissinger: Unification of the logic of causality
Employ the framework of (pre-)causal categories to unite notions of causality and techniques for causal reasoning which occur in classical statistics, quantum foundations, and beyond.
Reading: Kissinger and Uijlen; Henson, Lal, and Pusey.

Martha Lewis: Compositional approaches to linguistics and cognition
Use compact closed categories to integrate compositional models of meaning with distributed, geometric, and other meaning representations in linguistics and cognitive science.

Pawel Sobocinski: Modelling of open and interconnected systems
Use Carboni and Walters’ bicategories of relations as a multidisciplinary algebra of open and interconnected systems.

### Applications

We hope that each working group will comprise both participants who specialise in category theory and in the relevant application field. As a prerequisite, those participants specialising in category theory should feel comfortable with the material found in Categories for the Working Mathematician or its equivalent; those specialising in applications should have a similar graduate-level introduction.

To apply, please fill out the form here. You will be asked to upload a single PDF file containing the following information:

• Your contact information and educational history.
• A brief paragraph explaining your interest in this course.
• A paragraph or two describing one of your favorite topics in category theory, or your application field.
• A ranked list of the papers you would most like to present, together with an explanation of your preferences. Note that the paper you present determines which working group you will join.

Anyone is welcome to apply, although preference may be given to current graduate students and postdocs. Women and members of other underrepresented groups within applied category theory are particularly encouraged to apply.

Some support will be available to help with the costs (flights, accommodation, food, childcare) of attending the Research Week and the Workshop on Applied Category Theory; please indicate in your application if you would like to be considered for such support.

If you have any questions, please feel free to contact Brendan Fong (bfo at mit dot edu) or Nina Otter (otter at maths dot ox dot ac dot uk).

## Applied Category Theory at UCR (Part 2)

21 September, 2017

I’m running a special session on applied category theory, and now the program is available:

Applied category theory, Fall Western Sectional Meeting of the AMS, 4-5 November 2017, U.C. Riverside.

This is going to be fun.

My former student Brendan Fong is now working with David Spivak at MIT, and they’re both coming. My collaborator John Foley at Metron is also coming: we’re working on the CASCADE project for designing networked systems.

Dmitry Vagner is coming from Duke: he wrote a paper with David and Eugene Lerman on operads and open dynamical systems. Christina Vaisilakopoulou, who has worked with David and Patrick Schultz on dynamical systems, has just joined our group at UCR, so she will also be here. And the three of them have worked with Ryan Wisnesky on algebraic databases. Ryan will not be here, but his colleague Peter Gates will: together with David they have a startup called Categorical Informatics, which uses category theory to build sophisticated databases.

That’s not everyone—for example, most of my students will be speaking at this special session, and other people too—but that gives you a rough sense of some people involved. The conference is on a weekend, but John Foley and David Spivak and Brendan Fong and Dmitry Vagner are staying on for longer, so we’ll have some long conversations… and Brendan will explain decorated corelations in my Tuesday afternoon network theory seminar.

Here’s the program. Click on talk titles to see abstracts. For a multi-author talk, the person with the asterisk after their name is doing the talking. All the talks will be in Room 268 of the Highlander Union Building or ‘HUB’.

### Saturday November 4, 2017, 9:00 a.m.-10:50 a.m.

9:00 a.m.
A higher-order temporal logic for dynamical systems.
David I. Spivak, MIT

10:00 a.m.
Algebras of open dynamical systems on the operad of wiring diagrams.
Dmitry Vagner*, Duke University
David I. Spivak, MIT
Eugene Lerman, University of Illinois at Urbana-Champaign

10:30 a.m.
Abstract dynamical systems.
Christina Vasilakopoulou*, University of California, Riverside
David Spivak, MIT
Patrick Schultz, MIT

### Saturday November 4, 2017, 3:00 p.m.-5:50 p.m.

3:00 p.m.
Black boxes and decorated corelations.
Brendan Fong, MIT

4:00 p.m.
Compositional modelling of open reaction networks.
Blake S. Pollard*, University of California, Riverside
John C. Baez, University of California, Riverside

4:30 p.m.
A bicategory of coarse-grained Markov processes.
Kenny Courser, University of California, Riverside

5:00 p.m.
A bicategorical syntax for pure state qubit quantum mechanics.
Daniel M. Cicala, University of California, Riverside

5:30 p.m.
Open systems in classical mechanics.
Adam Yassine, University of California Riverside

### Sunday November 5, 2017, 9:00 a.m.-10:50 a.m.

9:00 a.m.
Controllability and observability: diagrams and duality.
Jason Erbele, Victor Valley College

9:30 a.m.
Frobenius monoids, weak bimonoids, and corelations.
Brandon Coya, University of California, Riverside

10:00 a.m.
Compositional design and tasking of networks.
John D. Foley*, Metron, Inc.
John C. Baez, University of California, Riverside
Joseph Moeller, University of California, Riverside
Blake S. Pollard, University of California, Riverside

10:30 a.m.
Joseph Moeller*, University of California, Riverside
John Foley, Metron Inc.
John C. Baez, University of California, Riverside
Blake S. Pollard, University of California, Riverside

### Sunday November 5, 2017, 2:00 p.m.-4:50 p.m.

2:00 p.m.
Reeb graph smoothing via cosheaves.
Vin de Silva, Department of Mathematics, Pomona College

3:00 p.m.
Knowledge representation in bicategories of relations.
Evan Patterson*, Stanford University, Statistics Department

3:30 p.m.
The multiresolution analysis of flow graphs.
Steve Huntsman*, BAE Systems

4:00 p.m.
Data modeling and integration using the open source tool Algebraic Query Language (AQL).
Peter Y. Gates*, Categorical Informatics
Ryan Wisnesky, Categorical Informatics

## Applied Category Theory 2018

12 September, 2017

There will be a conference on applied category theory!

Applied Category Theory (ACT 2018). School 23–27 April 2018 and workshop 30 April–4 May 2018 at the Lorentz Center in Leiden, the Netherlands. Organized by Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford).

The plenary speakers will be:

• Samson Abramsky (Oxford)
• John Baez (UC Riverside)
• Kathryn Hess (EPFL)
• David Spivak (MIT)

There will be a lot more to say as this progresses, but for now let me just quote from the conference website:

Applied Category Theory (ACT 2018) is a five-day workshop on applied category theory running from April 30 to May 4 at the Lorentz Center in Leiden, the Netherlands.

Towards an Integrative Science: in this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one scientific discipline can be reused in another. The aim of the workshop is to (1) explore the use of category theory within and across different disciplines, (2) create a more cohesive and collaborative ACT community, especially among early-stage researchers, and (3) accelerate research by outlining common goals and open problems for the field.

While the workshop will host discussions on a wide range of applications of category theory, there will be four special tracks on exciting new developments in the field:

1. Dynamical systems and networks
2. Systems biology
3. Cognition and AI
4. Causality

Accompanying the workshop will be an Adjoint Research School for early-career researchers. This will comprise a 16 week online seminar, followed by a 4 day research meeting at the Lorentz Center in the week prior to ACT 2018. Applications to the school will open prior to October 1, and are due November 1. Admissions will be notified by November 15.

Sincerely,
The organizers

Bob Coecke (Oxford), Brendan Fong (MIT), Aleks Kissinger (Nijmegen), Martha Lewis (Amsterdam), and Joshua Tan (Oxford)

Category theory is a branch of mathematics originally developed to transport ideas from one branch of mathematics to another, e.g. from topology to algebra. Applied category theory refers to efforts to transport the ideas of category theory from mathematics to other disciplines in science, engineering, and industry.

This site originated from discussions at the Computational Category Theory Workshop at NIST on Sept. 28-29, 2015. It serves to collect and disseminate research, resources, and tools for the development of applied category theory, and hosts a blog for those involved in its study.

### The proposal: Towards an Integrative Science

Category theory was developed in the 1940s to translate ideas from one field of mathematics, e.g. topology, to another field of mathematics, e.g. algebra. More recently, category theory has become an unexpectedly useful and economical tool for modeling a range of different disciplines, including programming language theory [10], quantum mechanics [2], systems biology [12], complex networks [5], database theory [7], and dynamical systems [14].

A category consists of a collection of objects together with a collection of maps between those objects, satisfying certain rules. Topologists and geometers use category theory to describe the passage from one mathematical structure to another, while category theorists are also interested in categories for their own sake. In computer science and physics, many types of categories (e.g. topoi or monoidal categories) are used to give a formal semantics of domain-specific phenomena (e.g. automata [3], or regular languages [11], or quantum protocols [2]). In the applied category theory community, a long-articulated vision understands categories as mathematical workspaces for the experimental sciences, similar to how they are used in topology and geometry [13]. This has proved true in certain fields, including computer science and mathematical physics, and we believe that these results can be extended in an exciting direction: we believe that category theory has the potential to bridge specific different fields, and moreover that developments in such fields (e.g. automata) can be transferred successfully into other fields (e.g. systems biology) through category theory. Already, for example, the categorical modeling of quantum processes has helped solve an important open problem in natural language processing [9].

In this workshop, we want to instigate a multi-disciplinary research program in which concepts, structures, and methods from one discipline can be reused in another. Tangibly and in the short-term, we will bring together people from different disciplines in order to write an expository survey paper that grounds the varied research in applied category theory and lays out the parameters of the research program.

In formulating this research program, we are motivated by recent successes where category theory was used to model a wide range of phenomena across many disciplines, e.g. open dynamical systems (including open Markov processes and open chemical reaction networks), entropy and relative entropy [6], and descriptions of computer hardware [8]. Several talks will address some of these new developments. But we are also motivated by an open problem in applied category theory, one which was observed at the most recent workshop in applied category theory (Dagstuhl, Germany, in 2015): “a weakness of semantics/CT is that the definitions play a key role. Having the right definitions makes the theorems trivial, which is the opposite of hard subjects where they have combinatorial proofs of theorems (and simple definitions). […] In general, the audience agrees that people see category theorists only as reconstructing the things they knew already, and that is a disadvantage, because we do not give them a good reason to care enough” [1, pg. 61].

In this workshop, we wish to articulate a natural response to the above: instead of treating the reconstruction as a weakness, we should treat the use of categorical concepts as a natural part of transferring and integrating knowledge across disciplines. The restructuring employed in applied category theory cuts through jargon, helping to elucidate common themes across disciplines. Indeed, the drive for a common language and comparison of similar structures in algebra and topology is what led to the development category theory in the first place, and recent hints show that this approach is not only useful between mathematical disciplines, but between scientific ones as well. For example, the ‘Rosetta Stone’ of Baez and Stay demonstrates how symmetric monoidal closed categories capture the common structure between logic, computation, and physics [4].

[1] Samson Abramsky, John C. Baez, Fabio Gadducci, and Viktor Winschel. Categorical methods at the crossroads. Report from Dagstuhl Perspectives Workshop 14182, 2014.

[2] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009.

[3] Michael A. Arbib and Ernest G. Manes. A categorist’s view of automata and systems. In Ernest G. Manes, editor, Category Theory Applied to Computation and Control. Springer, Berlin, 2005.

[4] John C. Baez and Mike STay. Physics, topology, logic and computation: a Rosetta stone. In Bob Coecke, editor, New Structures for Physics. Springer, Berlin, 2011.

[5] John C. Baez and Brendan Fong. A compositional framework for passive linear networks. arXiv e-prints, 2015.

[6] John C. Baez, Tobias Fritz, and Tom Leinster. A characterization of entropy in terms of information loss. Entropy, 13(11):1945–1957, 2011.

[7] Michael Fleming, Ryan Gunther, and Robert Rosebrugh. A database of categories. Journal of Symbolic Computing, 35(2):127–135, 2003.

[8] Dan R. Ghica and Achim Jung. Categorical semantics of digital circuits. In Ruzica Piskac and Muralidhar Talupur, editors, Proceedings of the 16th Conference on Formal Methods in Computer-Aided Design. Springer, Berlin, 2016.

[9] Dimitri Kartsaklis, Mehrnoosh Sadrzadeh, Stephen Pulman, and Bob Coecke. Reasoning about meaning in natural language with compact closed categories and Frobenius algebras. In Logic and Algebraic Structures in Quantum Computing and Information. Cambridge University Press, Cambridge, 2013.

[10] Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, 1991.

[11] Nicholas Pippenger. Regular languages and Stone duality. Theory of Computing Systems 30(2):121–134, 1997.

[12] Robert Rosen. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics, 20(4):317–341, 1958.

[13] David I. Spivak. Category Theory for Scientists. MIT Press, Cambridge MA, 2014.

[14] David I. Spivak, Christina Vasilakopoulou, and Patrick Schultz. Dynamical systems and sheaves. arXiv e-prints, 2016.

## Complex Adaptive Systems (Part 5)

4 September, 2017

When we design a complex system, we often start with a rough outline and fill in details later, one piece at a time. And if the system is supposed to be adaptive, these details may need to changed as the system is actually being used!

The use of operads should make this easier. One reason is that an operad typically has more than one algebra.

Remember from Part 3: an operad has operations, which are abstract ways of sticking things together. An algebra makes these operations concrete: it specifies some sets of actual things, and how the operations in the operad get implemented as actual ways to stick these things together.

So, an operad $O$ can have one algebra in which things are described in a bare-bones, simplified way, and another algebra in which things are described in more detail. Indeed it will typically have many algebras, corresponding to many levels of detail, but let’s just think about two for a minute.

When we have a ‘less detailed’ algebra $A$ and a ‘more detailed’ algebra $A',$ they will typically be related by a map

$f : A' \to A$

which ‘forgets the extra details’. This map should be a ‘homomorphism’ of algebras, but I’ll postpone the definition of that concept.

What we often want to do, when designing a system, is not forget extra detail, but rather add extra detail to some rough specification. There is not always a systematic way to do this. If there is, then we may have a homomorphism

$g : A \to A'$

going back the other way. This is wonderful, because it lets us automate the process of filling in the details. But we can’t always count on being able to do this—especially not if we want an optimal or even acceptable result. So, often we may have to start with an element of $A$ and search for elements of $A'$ that are mapped to it by $f : A' \to A.$

Let me give some examples. I’ll take the operad that I described last time, and describe some of its algebras, and homomorphisms between these.

I’ll start with an algebra that has very little detail: its elements will be simple graphs. As the name suggests, these are among the simplest possible ways of thinking about networks. They just look like this:

Then I’ll give an algebra with more detail, where the vertices of our simple graphs are points in the plane. There’s nothing special about the plane: we could replace the plane by any other set, and get another algebra of our operad. For example, we could use the set of points on the surface of the Caribbean Sea, the blue stuff in the rectangle here:

That’s what we might use in a search and rescue operation. The points could represent boats, and the edges could represent communication channels.

Then I’ll give an algebra with even more detail, where two points connected by an edge can’t be too far apart. This would be good for range-limited communication channels.

Then I’ll give an algebra with still more detail, where the locations of the points are functions of time. Now our boats are moving around!

Okay, here we go.

The operad from last time was called $O_G.$ Here $G$ is the network model of simple graphs. The best way to picture an operation of $O_G$ is as a way of sticking together a list of simple graphs to get a new simple graph.

For example, an operation

$f \in O_G(3,4,2;9)$

is a way of sticking together a simple graph with 3 vertices, one with 4 vertices and one with 2 vertices to get one with 9 vertices. Here’s a picture of such an operation:

Note that this operation is itself a simple graph. An operation in $O_G(3,4,2;9)$ is just a simple graph with 9 vertices, where we have labelled the vertices from 1 to 9.

This operad comes with a very obvious algebra $A$ where the operations do just what I suggested. In this algebra, an element of $A(t)$ is a simple graph with $t$ vertices, listed in order. Here $t$ is any natural number, which I’m calling ‘t’ for ‘type’.

We also need to say how the operations in $O_G$ act on these sets $A(t).$ If we take simple graphs in $A(3), A(4),$ and $A(2)$:

we can use our operation $f$ to stick them together and get this:

But we can also make up a more interesting algebra of $O_G.$ Let’s call this algebra $A'.$ We’ll let an element of $A'(t)$ be a simple graph with $t$ vertices, listed in order, which are points in the plane.

My previous pictures can be reused to show how operations in $O_G$ act on this new algebra $A'.$ The only difference is that now we tread the vertices literally as points in the plane! Before you should have been imagining them as abstract points not living anywhere; now they have locations.

Now let’s make up an even more detailed algebra $A''.$

What if our communication channels are ‘range-limited’? For example, what if two boats can’t communicate if they are more than 100 kilometers apart?

Then we can let an element of $A''(t)$ be a simple graph with $t$ vertices in the plane such that no two vertices connected by an edge have distance > 100.

Now the operations of our operad $O_G$ act in a more interesting way. If we have an operation, and we apply it to elements of our algebra, it ‘tries’ to put in new edges as it did before, but it ‘fails’ for any edge that would have length > 100. In other words, we just leave out any edges that would be too long.

It took me a while to figure this out. At first I thought the result of the operation would need to be undefined whenever we tried to create an edge that violated the length constraint. But in fact it acts in a perfectly well-defined way: we just don’t put in edges that would be too long!

This is good. This means that if you tell two boats to set up a communication channel, and they’re too far apart, you don’t get the ‘blue screen of death’: your setup doesn’t crash and burn. Instead, you just get a polite warning—‘communication channel not established’—and you can proceed.

The nontrivial part is to check that if we do this, we really get an algebra of our operad! There are some laws that must hold in any algebra. But since I haven’t yet described those laws, I won’t check them here. You’ll have to wait for our paper to come out.

Let’s do one more algebra today. For lack of creativity I’ll call it $A'''.$ Now an element of $A'''(t)$ is a time-dependent graph in the plane with $t$ vertices, listed in order. Namely, the positions of the vertices depend on time, and the presence or absence of an edge between two vertices can also depend on time. Furthermore, let’s impose the requirement that any two vertices can only connected by an edge at times when their distance is ≤ 100.

When I say ‘functions of time’ here, what ‘time’? We can model time by some interval $[T_1, T_2].$ But if you don’t like that, you can change it.

This algebra $A'''$ works more or less like $A''.$ The operations of $O_G$ try to create edges, but these edges only ‘take’ at times when the vertices they connect have distance ≤ 100.

There’s something here you might not like. Our operations can only try to create edges ‘for all times’… and succeed at times when the vertices are close enough. We can’t try to set up a communication channel for a limited amount of time.

But fear not: this is just a limitation in our chosen network model, ‘simple graphs’. With a fancier network model, we’d get a fancier operad, with fancier operations. Right now I’m trying to keep the operad simple (pun not intended), and show you a variety of different algebras.

And you might expect, we have algebra homomorphisms going from more detailed algebras to less detailed ones:

$f_T : A''' \to A'', \quad h : A' \to A$

The homomorphism $h$ takes a simple graph in the plane and forgets the location of its vertices. The homomorphism $f_T$ depends on a choice of time $T \in [T_1, T_2].$ For any time $T,$ it takes a time-dependent graph in the plane and evaluates it at that time, getting a graph in the plane (which obeys the distance constraints, since the time-dependent graph obeyed those constraints at any time).

We do not have a homomorphism $g: A'' \to A'$ that takes a simple graph in the plane obeying our distance constraints and forgets about those constraints. There’s a map $g$ sending elements of $A''$ to elements of $A'$ in this way. But it’s not an algebra homomorphism! The problem is that first trying to connect two graphs with an edge and then applying $g$ may give a different result than first applying $g$ and then connecting two graphs with an edge.

In short: a single operad has many algebras, which we can use to describe our desired system at different levels of detail. Algebra homomorphisms relate these different levels of detail.

Next time I’ll look at some more interesting algebras of the same operad. For example, there’s one that describes a system of interacting mobile agents, which move around in some specific way, determined by their location and the locations of the agents they’re communicating with.

Even this is just the tip of the iceberg—that is, still a rather low level of detail. We can also introduce stochasticity (that is, randomness). And to go even further, we could switch to a more sophisticated operad, based on a fancier ‘network model’.

But not today.

## Complex Adaptive System Design (Part 4)

22 August, 2017

Last time I introduced typed operads. A typed operad has a bunch of operations for putting together things of various types and getting new things of various types. This is a very general idea! But in the CASCADE project we’re interested in something more specific: networks. So we want operads whose operations are ways to put together networks and get new networks.

That’s what our team came up with: John Foley of Metron, my graduate students Blake Pollard and Joseph Moeller, and myself. We’re writing a couple of papers on this, and I’ll let you know when they’re ready. These blog articles are kind of sneak preview—and a gentle introduction, where you can ask questions.

For example: I’m talking a lot about networks. But what is a ‘network’, exactly?

There are many kinds. At the crudest level, we can model a network as a simple graph, which is something like this:

There are some restrictions on what counts as a simple graph. If the vertices are agents of some sort and the edges are communication channels, these restrictions imply:

• We allow at most one channel between any pair of agents, since there’s at most one edge between any two vertices of our graph.

• The channels do not have a favored direction, since there are no arrows on the edges of our graph.

• We don’t allow a channel from an agent to itself, since an edge can’t start and end at the same vertex.

For other purposes we may want to drop some or all of these restrictions. There is an appalling diversity of options! We might want to allow multiple channels between a pair of agents. For this we could use multigraphs. We might want to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. For this we could use directed graphs. And so on.

We will also want to consider graphs with colored vertices, to specify different types of agents—or colored edges, to specify different types of channels. Even more complicated variants are likely to become important as we proceed.

To avoid sinking into a mire of special cases, we need the full power of modern mathematics. Instead of separately studying all these various kinds of networks, we need a unified notion that subsumes all of them.

To do this, the Metron team came up with something called a ‘network model’. There is a network model for simple graphs, a network model for multigraphs, a network model for directed graphs, a network model for directed graphs with 3 colors of vertex and 15 colors of edge, and more.

You should think of a network model as a kind of network. Not a specific network, just a kind of network.

Our team proved that for each network model $G$ there is an operad $O_G$ whose operations describe how to put together networks of that kind. We call such operads ‘network operads’.

I want to make all this precise, but today let me just show you one example. Let’s take $G$ to be the network model for simple graphs, and look at the network operad $O_G.$ I won’t tell you what kind of thing $G$ is yet! But I’ll tell you about the operad $O_G$.

Types. Remember from last time that an operad has a set of ‘types’. For $O_G$ this is the set of natural numbers, $\mathbb{N}.$ The reason is that a simple graph can have any number of vertices.

Operations. Remember that an operad has sets of ‘operations’. In our case we have a set of operations $O_G(t_1,\dots,t_n ; t)$ for each choice of $t_1,\dots,t_n, t \in \mathbb{N}.$

An operation $f \in O_G(t_1,\dots,t_n; t)$ is a way of taking a simple graph with $t_1$ vertices, a simple graph with $t_2$ vertices,… and so on, and sticking them together, perhaps adding new edges, to get a simple graph with

$t = t_1 + \cdots + t_n$

vertices.

Let me show you an operation

$f \in O_G(3,4,2;9)$

This will be a way of taking three simple graphs—one with 3 vertices, one with 4, and one with 2—and sticking them together, perhaps adding edges, to get one with 9 vertices.

Here’s what $f$ looks like:

It’s a simple graph with vertices numbered from 1 to 9, with the vertices in bunches: {1,2,3}, {4,5,6,7} and {8,9}. It could be any such graph. This one happens to have an edge from 3 to 6 and an edge from 1 to 2.

Here’s how we can actually use our operation. Say we have three simple graphs like this:

Then we can use our operation to stick them together and get this:

Notice that we added a new edge from 3 to 6, connecting two of our three simple graphs. We also added an edge from 1 to 2… but this had no effect, since there was already an edge there! The reason is that simple graphs have at most one edge between vertices.

But what if we didn’t already have an edge from 1 to 2? What if we applied our operation $f$ to the following simple graphs?

Well, now we’d get this:

This time adding the edge from 1 to 2 had an effect, since there wasn’t already an edge there!

In short, we can use this operad to stick together simple graphs, but also to add new edges within the simple graphs we’re sticking together!

When I’m telling you how we ‘actually use’ our operad to stick together graphs, I’m secretly describing an algebra of our operad. Remember, an operad describes ways of sticking together things together, but an ‘algebra’ of the operad gives a particular specification of these things and describes how we stick them together.

Our operad $O_G$ has lots of interesting algebras, but I’ve just shown you the simplest one. More precisely:

Things. Remember from last time that for each type, an algebra specifies a set of things of that type. In this example our types are natural numbers, and for each natural number $t \in \mathbb{N}$ I’m letting the set of things $A(t)$ consist of all simple graphs with vertices $\{1, \dots, t\}.$

Action. Remember that our operad $O_G$ should have an action on $A$, meaning a bunch of maps

$\alpha : O_G(t_1,...,t_n ; t) \times A(t_1) \times \cdots \times A(t_n) \to A(t)$

I just described how this works in some examples. Some rules should hold… and they do.

To make sure you understand, try these puzzles:

Puzzle 1. In the example I just explained, what is the set $O_G(t_1,\dots,t_n ; t)$ if $t \ne t_1 + \cdots + t_n?$

Puzzle 2. In this example, how many elements does $O_G(1,1;2)$ have?

Puzzle 3. In this example, how many elements does $O_G(1,2;3)$ have?

Puzzle 4. In this example, how many elements does $O_G(1,1,1;3)$ have?

Puzzle 5. In the particular algebra $A$ that I explained, how many elements does $A(3)$ have?

Next time I’ll describe some more interesting algebras of this operad $O_G.$ These let us describe networks of mobile agents with range-limited communication channels!