## 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:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/April.2022nd.3A.20Michael.20Shulman

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.

## Open Games: the Long Road to Practical Applications

13 April, 2020

In the third talk of the ACT@UCR seminar, Jules Hedges spoke about open games!

He gave his talk on Wednesday April 15th. Afterwards we discussed it at the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/

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:

• April 15, Jules Hedges: Open games: the long road to practical applications.

Abstract. I will talk about open games, and the closely related concepts of lenses/optics and open learners. My goal is to report on the successes and failures of an ongoing effort to try to realise the often-claimed benefits of categories and compositionality in actual practice. I will introduce what little theory is needed along the way. Here are some things I plan to talk about:

— Lenses as an abstraction of the chain rule

— Comb diagrams

— Surprising applications of open games: Bayesian inference, value function iteration

— The state of tool support

— Open games in their natural habitat: microeconomics

— Sociological aspects of working with economics

## Mathematics Seminars List

9 April, 2020

The coronavirus pandemic is pushing us to take seriously a revolutionary new technology called the “internet”. Here’s a nice list of online math talks, organized by Jaume de Dios:

(Click for more on this picture.)

## Category Theory Calendar

6 April, 2020

There are now enough online events in category theory that a calendar is needed. And here it is!

It should show the times in your time zone, at least if you don’t prevent it from getting that information.

## 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.

## A Categorical View of Conditional Expectation

2 April, 2020

$\;$

I always like to see categories combined with probability theory and analysis. So I’m glad Prakash Panangaden did that in his talk at the ACT@UCR seminar.

He gave his talk on Wednesday April 8th. Afterwards we had discussions 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:

A categorical view of conditional expectation

Abstract. This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones—which are abstract versions of the familiar cones in vector spaces—of measures and related categories cones of Lp functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

For more, see:

• Philippe Chaput, Vincent Danos, Prakash Panangaden and Gordon Plotkin, Approximating Markov processes by averaging, in International Colloquium on Automata, Languages, and Programming, Springer, Berlin, 2009.

## Structured Cospans and Double Categories

31 March, 2020

$\; \phantom{a}$

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:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/April.201st.3A.20John.20Baez

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:

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.

## Online Worldwide Seminar on Logic and Semantics

30 March, 2020

Someone should make a grand calendar, readable by everyone, of all the new math seminars that are springing into existence. Here’s another! It’s a bit outside the core concerns of Azimuth, but it’ll have a lot of category theory, and it features some good practices that I hope more seminars adopt, or tweak.

Online Worldwide Seminar on Logic and Semantics, organized by Alexandra Silva, Pawel Sobocinski and Jamie Vicary.

There will be talks fortnightly at 1 pm UTC, which is currently 2 pm British Time, thanks to daylight savings time. Here are the first few:

• Wednesday, April 1, — Kevin Buzzard, Imperial College London: “Is HoTT the way to do mathematics?”

• Wednesday, April 15 — Joost-Pieter Katoen, Aachen University: “Termination of probabilistic programs”.

• Wednesday, April 29 — Daniela Petrisan, University of Paris: “Combining probabilistic and non-deterministic choice via weak distributive laws”.

• Wednesday, May 13 — Bartek Klin, Warsaw University: “Monadic monadic second order logic”.

• Wednesday, May 27 — Dexter Kozen, Cornell University: “Brzozowski derivatives as distributive laws”.

Joining the seminar.To join any OWLS seminar, visit the following link, up to 15 minutes before the posted start time:

You will be given the option to join through your web browser, or to launch the Zoom client if it is installed on your device. For the best experience, we recommend using the client.

Audio and video. We encourage all participants to enable their audio and video at all times (click “Use Device Audio” in the Zoom interface.) Don’t worry about making noise and disrupting the proceedings accidentally; the Chairperson will ensure your audio is muted by default during the seminar. Having your audio and video enabled will allow other participants to see your face in the “Gallery” view, letting them know that you’re taking part. It also gives you the option of asking a question, and of making best use of the “coffee break” sessions. For most users with good network access (such as a fast home broadband connection), there is no need to worry that having your audio and video enabled will degrade the experience; the technology platform ensures that the speaker’s audio/video stream is prioritised at all times. However, those on slow connections may find it better to disable their audio and video.

Coffee breaks. Every OWLS seminar has two “coffee breaks”, one starting 15 minutes before the posted start time of the seminar, and the second starting after the seminar is finished. To participate in these, feel free to join the meeting early, or to keep the meeting window open after the end of the talk. During these coffee break periods, participants will be automatically gathered into small groups, assigned at random; please introduce yourself to the other members of your group, and chat just like you would at a real conference. Remember to bring your own coffee!

During the seminar. If you’d like to ask a question, either during the seminar or in the question period at the end, click the “Participants” menu and select “Raise hand”. The Chairperson may choose to interrupt the speaker and give your audio/video feed the focus, giving you the opportunity to ask your question verbally, or may instead decide to let the seminar continue. You may click “Lower hand” at any time to show you no longer wish to ask a question. To preserve the experience of a real face-to-face conference, there is no possibility of giving a written question, and the chat room is disabled at all times. You also have the opportunity to give nonverbal feedback to the speaker by clicking the “speed up” or “slow down” buttons, also in the “Participants” menu.

Recordings. All OWLS seminars are recorded and uploaded to YouTube after the event. Only the audio/video of the chairperson, speaker, and questioners will be captured. If you prefer not to be recorded, do not ask a question. Of course, the organizers do not make any recordings of the coffee break sessions.

## MIT Categories Seminar

25 March, 2020

The MIT Categories Seminar is an informal teaching seminar in category theory and its applications, with the occasional research talk. This spring they are meeting online each Thursday, 12 noon to 1pm Eastern Time.

The talks are broadcast over YouTube here, with simultaneous discussion on the Category Theory Community Server. (To join the channel, click here.) Talks are recorded and remain available on YouTube.

Here are some forthcoming talks:

March 26: David Jaz Myers (Johns Hopkins University) — Homotopy type theory for doing category theory.

April 2: Todd Trimble (Western Connecticut State University) — Geometry of regular relational calculus.

April 9: John Baez (UC Riverside) — Structured cospans and Petri nets.

April 16: Joachim Kock (Universitat Autònoma de Barcelona) — to be announced.

April 23: Joe Moeller (UC Riverside) — to be announced.

Videos of many older talks are available here.

## ACT@UCR Seminar

24 March, 2020

We’re having a seminar on applied category theory at U. C. Riverside, organized by Joe Moeller and Christian Williams.

It will take place on Wednesdays at 5 pm UTC, which is 10 am in California or 1 pm on the east coast of the United States, or 6 pm in England. It will be held online via Zoom, here:

https://ucr.zoom.us/j/607160601

We’ll have discussions on the Category Theory Community Server.

All talks will be recorded and put on YouTube here:

Here are the talks:

• April 1st, John Baez: 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

L(a) → x ← L(b).

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.

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

The talk was based on these papers:

• John Baez and Kenny Courser, Structured cospans.
• Kenny Courser, Open Systems: a Double Categorical
Perspective
.

• April 8th, Prakash Panangaden: A categorical view of conditional expectation.

Abstract. This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones—which are abstract versions of the familiar cones in vector spaces—of measures and related categories cones of Lp functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin.

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

His talk was based on this paper:

• April 15, Jules Hedges: Open games: the long road to practical applications.

Abstract. I will talk about open games, and the closely related concepts of lenses/optics and open learners. My goal is to report on the successes and failures of an ongoing effort to try to realise the often-claimed benefits of categories and compositionality in actual practice. I will introduce what little theory is needed along the way. Here are some things I plan to talk about:

— Lenses as an abstraction of the chain rule

— Comb diagrams

— Surprising applications of open games: Bayesian inference, value function iteration

— The state of tool support

— Open games in their natural habitat: microeconomics

— Sociological aspects of working with economics

You can see the slides here, or download a video of the 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.

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

His talk is based on this paper:

• April 29, 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.

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

• May 6, Sarah Rovner-Frydman: Separation logic through a new lens.

Abstract. Separation logic aims to reason compositionally about the behavior of programs that manipulate shared resources. When working with separation logic, it is often necessary to manipulate information about program state in patterns of deconstruction and reconstruction. This achieves a kind of “focusing” effect which is somewhat reminiscent of using optics in a functional programming language. We make this analogy precise by showing that several interrelated techniques in the literature for managing these patterns of manipulation can be derived as instances of the general definition of profunctor optics. In doing so, we specialize the machinery of profunctor optics from categories to posets and to sets. This simplification makes most of this machinery look more familiar, and it reveals that much of it was already hiding in separation logic in plain sight.

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

• May 13, Tai-Danae Bradley: Formal concepts vs. eigenvectors of density operators.

Abstract. In this talk, I’ll show how any probability distribution on a product of finite sets gives rise to a pair of linear maps called density operators, whose eigenvectors capture “concepts” inherent in the original probability distribution. In some cases, the eigenvectors coincide with a simple construction from lattice theory known as a formal concept. In general, the operators recover marginal probabilities on their diagonals, and the information stored in their eigenvectors is akin to conditional probability. This is useful in an applied setting, where the eigenvectors and eigenvalues can be glued together to reconstruct joint probabilities. This naturally leads to a tensor network model of the original distribution. I’ll explain these ideas from the ground up, starting with an introduction to formal concepts. Time permitting, I’ll also share how the same ideas lead to a simple framework for modeling hierarchy in natural language. As an aside, it’s known that formal concepts arise as an enriched version of a generalization of the Isbell completion of a category. Oftentimes, the construction is motivated by drawing an analogy with elementary linear algebra. I like to think of this talk as an application of the linear algebraic side of that analogy.

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

Her talk is based on her thesis:

• May 20, Gordon Plotkin: A complete axiomatisation of partial differentiation

Abstract. We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof makes use of Severi’s theorem that all multivariate Hermite problems are solvable. We also hope to present a number of related results, such as decidability and Hilbert-Post completeness.

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

• May 27, Simon Willerton: The Legendre–Fenchel transform from a category theoretic perspective.

Abstract. The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I’ll show how it arises in the context of category theory using categories enriched over the extended real numbers $\overline{ \mathbb{R}}:=[-\infty,+\infty]$. It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (e.g. in Galois theory or algebraic geometry) arise from a relation between sets.

I won’t assume knowledge of the Legendre-Fenchel transform.

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

• June 3, Nina Otter: Values and inclusivity in the applied category theory community.

Abstract. Saddened by the current events, we are taking this opportunity to pause and reflect on what we can do to change the status quo and try to bring about real and long-lasting change. Thus, we are holding a discussion aimed at finding concrete solutions to make the Applied Category Theory community more inclusive, and also to reflect about the values that our community would like to stand for and endorse, in particular, in terms of which sources of funding go against our values. While this discussion is specific to the applied category theory community, we believe that many of the topics will be of interest also to people in other fields, and thus we welcome anybody with an interest to attend. The discussion will consist of two parts: we will have first several people give short talks to discuss common issues that we need to address, as well as present specific plans for initiatives that we could take. We believe that the current pandemic, and the fact that all activities are now taking place remotely, gives us the opportunity to involve people who would otherwise find it difficult to travel, because of disabilities, financial reasons or care-taking responsibilities. Thus, now we have the opportunity to come up with new types of mentoring, collaborations, and many other initiatives that might have been difficult to envision until just a couple of months ago. The second part of the discussion will take place on the category theory community server, and its purpose is to allow for a broader participation in the discussion, and ideally during this part we will be able to flesh out in detail the specific initiatives that have been proposed in the talks.

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