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:


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.

The Monoidal Grothendieck Construction

24 April, 2020

My student Joe Moeller gave a talk at the MIT Categories Seminar today! People discussed his talk at the Category Theory Community Server, and if you join that you can see the discussion here:


You can see his slides here, and watch a video of his talk here:

The monoidal Grothendieck construction

Abstract. The Grothendieck construction gives an equivalence between fibrations and indexed categories. We will begin with a review of the classical story. We will then lift this correspondence to two monoidal variants, a global version and a fibre-wise version. Under certain conditions these are equivalent, so one can transfer fibre-wise monoidal structures to the total category. We will give some examples demonstrating the utility of this construction in applied category theory and categorical algebra.

The talk is based on this paper:

• Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck construction.

This, in turn, had its roots in our work on network models, a setup for the compositional design of networked systems:

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

Star-Autonomous Envelopes

21 April, 2020


In the fourth talk of the ACT@UCR seminar, Michael Shulman told us how to create nice string diagams for any closed symmetric monoidal category.

Mike had to teach right after his talk, but he rejoined us for discussions later at the Category Theory Community Server, here:


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

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

• April 22, Michael Shulman, Star-autonomous envelopes.

Abstract. Symmetric monoidal categories with duals, a.k.a. compact monoidal categories, have a pleasing string diagram calculus. In particular, any compact monoidal category is closed with [A,B] = (A* ⊗ B), and the transpose of A ⊗ B → C to A → [B,C] is represented by simply bending a string. Unfortunately, a closed symmetric monoidal category cannot even be embedded fully-faithfully into a compact one unless it is traced; and while string diagram calculi for closed monoidal categories have been proposed, they are more complicated, e.g. with “clasps” and “bubbles”. In this talk we obtain a string diagram calculus for closed symmetric monoidal categories that looks almost like the compact case, by fully embedding any such category in a star-autonomous one (via a functor that preserves the closed structure) and using the known string diagram calculus for star-autonomous categories. No knowledge of star-autonomous categories will be assumed.

His talk is based on this paper:

• Michael Shulman, Star-autonomous envelopes.

This subject is especially interesting to me since Mike Stay and I introduced string diagrams for closed monoidal categories in a somewhat ad hoc way in our Rosetta Stone paper—but the resulting diagrams required clasps and bubbles:


This is the string diagram for beta reduction in the cartesian closed category coming from the lambda calculus.

Structured Cospans and Petri Nets

6 April, 2020


This talk on structured cospans and Petri nets is the second of a two-part series, but it should be understandable on its own. The first part is on structured cospans and double categories.

I gave this second talk at the MIT Categories Seminar. People discussed the talk at the Category Theory Community Server.

You can see the slides here and watch a video here:

Structured cospans and Petri nets

Abstract. “Structured cospans” are a general way to study networks with inputs and outputs. Here we illustrate this using a type of network popular in theoretical computer science: Petri nets. An “open” Petri net is one with certain places designated as inputs and outputs. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Using the formalism of structured cospans, open Petri nets can be treated as morphisms of a symmetric monoidal category—or better, a symmetric monoidal double category. We explain two forms of semantics for open Petri nets using symmetric monoidal double functors out of this double category. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. The second, a “reachability” semantics, simply says what these processes can accomplish. This is joint work with Kenny Courser and Jade Master.

The talk is based on these papers:

• John Baez and Kenny Courser, Structured cospans.

• John Baez and Jade Master, Open Petri nets.

• Jade Master, Generalized Petri nets.

I’ve blogged about open Petri nets before, and these articles might be a good way to start learning about them:

Part 1: the double category of open Petri nets.

Part 2: the reachability semantics for open Petri nets.

Part 3: the free symmetric monoidal category on a Petri net.

Structured Cospans and Double Categories

31 March, 2020

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This talk on structured cospans and double categories is the first of a two-part series; the second part is about structured cospans and Petri nets.

I gave the first talk at the ACT@UCR seminar, on Wednesday April 1st. Afterwards we discussed it on the Category Theory Community Server, here:


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

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

Structured cospans and double categories

Abstract. One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce “structured cospans” as a way to study networks with inputs and outputs. Given a functor L: A → X, a structured cospan is a diagram in X of the form

If A and X have finite colimits and L is a left adjoint, we obtain a symmetric monoidal category whose objects are those of A and whose morphisms are certain equivalence classes of structured cospans. However, this arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans, not equivalence classes thereof. We explain the mathematics and illustrate it with an example from epidemiology.

This talk was based on work with Kenny Courser and Christina Vasilakopoulou, some of which appears here:

• John Baez and Kenny Courser, Structured cospans.

• Kenny Courser, Open Systems: a Double Categorical Perspective.

Yesterday Rongmin Lu told me something amazing: structured cospans were invented in 2007 by José Luiz Fiadeiro and Vincent Schmit. It’s pretty common for simple ideas to be discovered several times. The amazing thing is that these other authors also called these things ‘structured cospans’!

• José Luiz Fiadeiro and Vincent Schmitt, Structured co-spans: an algebra of interaction protocols, in International Conference on Algebra and Coalgebra in Computer Science, Springer, Berlin, 2007.

These earlier authors did not do everything we’ve done, so I’m not upset. Their work proves I chose the right name.

Applied Category Theory 2020 (Part 2)

23 March, 2020

Due to the coronavirus outbreak, many universities are moving activities online. This is a great opportunity to open up ACT2020 to a broader audience, with speakers from around the world.

The conference will take place July 6-10 online, coordinated by organizers in Boston USA. Each day there will be around six hours of live talks, which will be a bit more spaced out than usual to accommodate the different time zones of our speakers. All the talks will be both live streamed and recorded on YouTube. We will also have chat rooms and video chats in which participants can discuss various themes in applied category theory.

We will give more details as they become available and post updates on our official webpage:


Since there is no need to book travel, we were able to postpone the acceptance notification, and hence the submission deadline. If you would like to speak, please prepare an abstract or a conference paper according to the instructions here:


Important dates (all in 2020)

• Submission of contributed papers: May 10
• Acceptance/Rejection notification: June 7
• Tutorial day: July 5
• Main conference: July 6-10

Registration will now be free; please register for the conference ahead of time here:


We will send registering participants links to the live stream, the recordings, and the chat rooms, and we’ll use the list to inform participants of any changes.


To give a talk at ACT2020, you have to submit a paper. You can submit either original research papers or extended abstracts of work submitted/accepted/published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume.

Here’s how to submit papers. Two types of submissions are accepted, which will be reviewed to the same standards:

Proceedings Track. Original contributions of high quality work consisting of a 5–12 page extended abstract that provides evidence for results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submissions of works in progress are encouraged, but must be more substantial than a research proposal.

Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure these pages contain sufficient evidence of the quality and rigor of your work.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Submission is currently open, and can be perfomed at the following web page:


One or more best paper awards may be given out at the discretion of the PC chairs. Selected contributions will be offered extended keynote slots in the program.


Here are the local organizers:

• Destiny Chen (administration)
• Brendan Fong
• David Jaz Myers (logistics)
• Paolo Perrone (publicity)
• David Spivak

Here is the committee running the school:

• Carmen Constantin
• Eliana Lorch
• Paolo Perrone

Here is the steering committee:

• John Baez
• Bob Coecke
• David Spivak
• Christina Vasilakopoulou

Here is the program committee:

• Mathieu Anel, CMU
• John Baez, University of California, Riverside
• Richard Blute, University of Ottawa
• Tai-Danae Bradley, City University of New York
• Andrea Censi, ETC Zurich
• Bob Coecke, University of Oxford
• Valeria de Paiva, Samsung Research America and University of Birmingham
• Ross Duncan, University of Strathclyde
• Eric Finster, University of Birmingham
• Brendan Fong, Massachusetts Institute of Technology
• Tobias Fritz, Perimeter Institute for Theoretical Physics
• Richard Garner, Macquarie University
• Fabrizio Romano Genovese, Statebox
• Amar Hadzihasanovic, IRIF, Université de Paris
• Helle Hvid Hansen, Delft University of Technology
• Jules Hedges, Max Planck Institute for Mathematics in the Sciences
• Kathryn Hess Bellwald, Ecole Polytechnique Fédérale de Lausanne
• Chris Heunen, The University of Edinburgh
• Joachim Kock, UAB
• Tom Leinster, The University of Edinburgh
• Martha Lewis, University of Amsterdam
• Daniel R. Licata, Wesleyan University
• David Jaz Myers, Johns Hopkins University
• Paolo Perrone, MIT
• Vaughan Pratt, Stanford University
• Peter Selinger, Dalhousie University
• Michael Shulman, University of San Diego
David I. Spivak, MIT (co-chair)
• Walter Tholen, York University
• Todd Trimble, Western Connecticut State University
Jamie Vicary, University of Birmingham (co-chair)
• Maaike Zwart, University of Oxford

Applied Category Theory at NIST (Part 3)

22 February, 2020

Sadly, this workshop has been cancelled due to the coronavirus pandemic. It may be postponed to a later date.

My former student Blake Pollard is working at the National Institute of Standards and Technology. He’s working with Spencer Breiner and Eswaran Subrahmanian, who are big advocates of using category theory to organize design and manufacturing processes. In the spring of 2018 they had a workshop on applied category theory with a lot of honchos from industry and government in attendance—you can see videos by clicking the link.

This spring they’re having another workshop on this topic!

Applied Category Theory Workshop, April 8-9, 2020, National Institute of Standards and Technology, Gaithersburg, Maryland. Organized by Spencer Breiner, Blake Pollard and Eswaran Subrahmanian.

The focus of this workshop in on fostering the development of tooling and use-cases supporting the applied category theory community. We are particularly interested in bringing together practitioners who are engaged with susceptible domains as well as those involved in the implementation, support, and utilization of software and other tools. There will be a number of talks/demos showcasing existing approaches as well as ample time for discussion.

Here are the speakers listed so far:

• John Baez, University of California, Riverside

• Arquimedes Canedo, Siemens

• Daniel Cicala, New Haven University

• James Fairbanks, Georgia Tech Research Institute

• Jules Hedges, Max Planck Institute for the Mathematical Sciences

• Jelle Herold, Statebox

• Evan Patterson, Stanford University

• Qunfen Qi, University of Huddersfield

• Christian Williams, University of California, Riverside

• Ryan Wisnesky, Conexus.ai

I’ll also be giving a separate talk on “ecotechnology” at NIST on Friday April 10th; more about that later!