Graph Transformation Theory and Applications

4 December, 2020

I love graph rewriting—the study of ways to change one graph into another by changing one small part at a time. My student Daniel Cicala did his thesis on this! So I’m happy to hear about the new virtual seminar series GReTA: Graph TRansformation Theory and Applications.

It aims to serve as a platform for the international graph rewriting community, promote recent developments and trends in the field, and encourage regular networking and interaction between members of this community.

Seminars are held twice a month in the form of Zoom sessions (some of which will be live-streamed to YouTube). Go to the link if you want to join on Zoom.

• Subscribe to the GReTA YouTube channel.

• Subscribe to the GReTA Google Calendar (or alternatively import it in iCal format).

• Subscribe to the GReTA mailing list.

Here are the two talks so far. Any subject that can promote talks on both logic and chemistry must be good! Thinking of chemistry and logic as two aspects of the same thing is bound to trigger new ideas. (Just as a sequence of chemical reactions converts reactants into products, a proof converts assumptions into conclusions.)

Speaker: Barbara König
Title: Graph transformation meets logic

Abstract. We review the integration of (first-order) logic respectively nested conditions into graph transformation. Conditions can serve various purposes: they can constrain graph rewriting, symbolically specify sets of graphs, be used in query languages and in verification (for instance in Hoare logic and for behavioural equivalence checking). In the graph transformation community the formalism of nested graph conditions has emerged, that is, conditions which are equivalent to first-order logic, but directly integrate graphs and graph morphisms, in order to express constraints more succinctly. In this talk we also explain how the notion of nested conditions can be lifted from graph transformation systems to the setting of reactive systems as defined by Leifer and Milner. It turns out that some constructions for graph transformation systems (such as computing weakest preconditions and strongest postconditions and showing local confluence by means of critical pair analysis) can be done quite elegantly in the more general setting.

Speakers: Daniel Merkle and Jakob Lykke Andersen
Title: Chemical graph transformation and applications

Abstract: Any computational method in chemistry must choose some level of precision in the modeling. One choice is made in the methods of quantum chemistry based on quantum field theory. While highly accurate, the methods are computationally very demanding, which restricts their practical use to single reactions of molecules of moderate size even when run on supercomputers. At the same time, most existing computational methods for systems chemistry and biology are formulated at the other abstraction extreme, in which the structure of molecules is represented either not at all or in a very rudimentary fashion that does not permit the tracking of individual atoms across a series of reactions.

In this talk, we present our on-going work on creating a practical modelling framework for chemistry based on Double Pushout graph transformation, and how it can be applied to analyse chemical systems. We will address important technical design decisions as well as the importance of methods inspired from Algorithm Engineering in order to reach the required efficiency of our implementation. We will present chemically relevant features that our framework provides (e.g. automatic atom tracing) as well as a set of chemical systems we investigated are currently investigating. If time allows we will discuss variations of graph transformation rule compositions and their chemical validity.

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:

ACT2020 Program

27 June, 2020

Applied Category Theory 2020 is coming up soon! After the Tutorial Day on Sunday July 6th, there will be talks from Monday July 7th to Friday July 10th. All talks will be live on Zoom and on YouTube. Recorded versions will appear on YouTube later.

Here are the talks! They come in three kinds: keynotes, regular presentations and short industry presentations. Within each I’ve listed them in alphabetical order by speaker: I believe the first author is the speaker.

This is gonna be fun.

Keynote presentations (35 minutes)

• Henry Adams, Johnathan Bush and Joshua Mirth, Operations on metric thickenings.

• Nicolas Blanco and Noam Zeilberger: Bifibrations of polycategories and classical linear logic.

• Bryce Clarke, Derek Elkins, Jeremy Gibbons, Fosco Loregian, Bartosz Milewski, Emily Pillmore and Mario Román: Profunctor optics, a categorical update.

• Tobias Fritz, Tomáš Gonda, Paolo Perrone and Eigil Rischel: Distribution functors, second-order stochastic dominance and the Blackwell–Sherman–Stein Theorem in categorical probability.

• Micah Halter, Evan Patterson, Andrew Baas and James Fairbanks: Compositional scientific computing with Catlab and SemanticModels.

• Joachim Kock: Whole-grain Petri nets and processes.

• Andre Kornell, Bert Lindenhovius and Michael Mislove: Quantum CPOs.

• Martha Lewis: Towards logical negation in compositional distributional semantics.

• Jade Master and John Baez: Open Petri nets.

• Lachlan McPheat, Mehrnoosh Sadrzadeh, Hadi Wazni and Gijs Wijnholds, Categorical vector space semantics for Lambek calculus with a relevant modality.

• David Jaz Myers: Double categories of open dynamical systems.

• Toby St Clere Smithe, Cyber Kittens, or first steps towards categorical cybernetics.

Regular presentations (20 minutes)

• Robert Atkey, Bruno Gavranović, Neil Ghani, Clemens Kupke, Jeremy Ledent and Fredrik Nordvall Forsberg: Compositional game theory, compositionally.

• John Baez and Kenny Courser: Coarse-graining open Markov processes.

• Georgios Bakirtzis, Christina Vasilakopoulou and Cody Fleming, Compositional cyber-physical systems modeling.

• Marco Benini, Marco Perin, Alexander Alexander Schenkel and Lukas Woike: Categorification of algebraic quantum field theories.

• Daniel Cicala: Rewriting structured cospans.

• Bryce Clarke: A diagrammatic approach to symmetric lenses.

• Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, Alexis Toumi, Stefano Gogioso and Nicolo Chiappori: Quantum natural language processing.

• Geoffrey Cruttwell, Jonathan Gallagher and Dorette Pronk: Categorical semantics of a simple differential programming language.

• Swaraj Dash and Sam Staton: A monad for probabilistic point processes.

• Giovanni de Felice, Elena Di Lavore, Mario Román and Alexis Toumi: Functorial language games for question answering.

• Giovanni de Felice, Alexis Toumi and Bob Coecke: DisCoPy: monoidal categories in Python.

• Brendan Fong, David Jaz Myers and David I. Spivak: Behavioral mereology: a modal logic for passing constraints.

• Rocco Gangle, Gianluca Caterina and Fernando Tohme, A generic figures reconstruction of Peirce’s existential graphs (alpha).

• Jules Hedges and Philipp Zahn: Open games in practice.

• Jules Hedges: Non-compositionality in categorical systems theory.

• Michael Johnson and Robert Rosebrugh, The more legs the merrier: A new composition for symmetric (multi-)lenses.

• Joe Moeller, John Baez and John Foley: Petri nets with catalysts.

• John Nolan and Spencer Breiner, Symmetric monoidal categories with attributes.

• Joseph Razavi and Andrea Schalk: Gandy machines made easy via category theory.

• Callum Reader: Measures and enriched categories.

• Mario Román: Open diagrams via coend calculus.

• Luigi Santocanale, Dualizing sup-preserving endomaps of a complete lattice.

• Dan Shiebler: Categorical stochastic processes and likelihood.

• Richard Statman, Products in a category with only one object.

• David I. Spivak: Poly: An abundant categorical setting for mode-dependent dynamics.

• Christine Tasson and Martin Hyland, The linear-non-linear substitution 2-monad.

• Tarmo Uustalu, Niccolò Veltri and Noam Zeilberger: Proof theory of partially normal skew monoidal categories.

• Dmitry Vagner, David I. Spivak and Evan Patterson: Wiring diagrams as normal forms for computing in symmetric monoidal categories.

• Matthew Wilson, James Hefford, Guillaume Boisseau and Vincent Wang: The safari of update structures: visiting the lens and quantum enclosures.

• Paul Wilson and Fabio Zanasi: Reverse derivative ascent: a categorical approach to learning Boolean circuits.

• Gioele Zardini, David I. Spivak, Andrea Censi and Emilio Frazzoli: A compositional sheaf-theoretic framework for event-based systems.

Industry presentations (8 minutes)

• Arquimedes Canedo (Siemens Corporate Technology).

• Brendan Fong (Topos Institute).

• Jelle Herold (Statebox): Industrial strength CT.

• Steve Huntsman (BAE): Inhabiting the value proposition for category theory.

• Ilyas Khan (Cambridge Quantum Computing).

• Alan Ransil (Protocol Labs): Compositional data structures for the decentralized web.

• Alberto Speranzon (Honeywell).

• Ryan Wisnesky (Conexus): Categorical informatics at scale.

ACT2020 Tutorial Day

17 June, 2020

If you’re wanting to learn some applied category theory, register for the tutorials that are taking place on July 5, 2020 as part of ACT2020!

Applied category theory offers a rigorous mathematical language and toolset for relating different concepts from across math, science, and technology. For example, category theory finds common patterns between geometry (shapes), algebra (equations), numbers, logic, probability, etc. Applied category theory (ACT) looks for how those very same patterns extend outward to data, programs, processes, physics, linguistics, and so on—things we see in the real world. The field is currently growing, as new applications and common patterns are being found all the time. When you understand these ideas, more of your intuitions about the world can be made rigorous and thus be communicated at a larger scale. This in turn gives our community a chance to solve larger and more complex scientific, technological, and maybe even societal problems.

This year’s international applied category theory conference ACT2020 is having a tutorial day, meant to introduce newcomers to applied category theory. Tutorial day will take place on July 5 and will include a few main topics that will be taught semi-traditionally (via presentation, exercises, and discussion) over Zoom, as well as mentors who will be available throughout the day to work with smaller groups and/or individuals. We invite you to sign up here if you’re interested, so we can keep you posted. Hope to see you there!

The four courses will be roughly as follows:

• David Spivak: categorical databases for introducing sets, functions, categories, and functors.

• Fabrizio Genovese: string diagrams as a graphical language for category theory.

• Emily Riehl: the Yoneda lemma in the context of matrices.

A Complete Axiomatisation of Partial Differentiation

18 May, 2020

In the eighth talk of the ACT@UCR seminar, Gordon Plotkin told us about partial differentiation, viewed as a logical theory.

He gave his talk on Wednesday May 20th. Afterwards we discussed it on the Category Theory Community Server, here:

https://categorytheory.zulipchat.com/#narrow/stream/229966-ACT.40UCR-seminar/topic/May.2020th.3A.20Gordon.20Plotkin

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 his video here:

• 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.

The Category Theory Behind UMAP

10 February, 2020

An interesting situation has arisen. Some people working on applied category theory have been seeking a ‘killer app’: that is, an application of category theory to practical tasks that would be so compelling it would force the world to admit categories are useful. Meanwhile, the UMAP algorithm, based to some extent on category theory, has become very important in genomics:

• Leland McInnes, John Healy and James Melville, UMAP: uniform manifold approximation and projection for dimension reduction.

But while practitioners have embraced the algorithm, they’re still puzzled by its category-theoretic underpinnings, which are discussed in Section 2 of the paper. (You can read the remaining sections, which describe the algorithm quite concretely, without understanding Section 2.)

I first heard of this situation on Twitter when James Nichols wrote:

Wow! My first sighting of applied category theory: the UMAP algorithm. I’m a category novice, but the resulting adjacency-graph algorithm is v simple, so surely the theory boils down to reasonably simple arguments in topology/Riemannian geometry?

Do any of you prolific ACT tweeters know much about UMAP? I understand the gist of the linked paper, but not say why we need category theory to define this “fuzzy topology” concept, as opposed to some other analytic defn.

What was gained by CT for UMAP? (honest question, not trying to be snarky)

Leland McInnes, one of the inventors of UMAP, responded:

It is my math background, how I think about the problem, and how the algorithm was derived. It wasn’t something that was added, but rather something that was always there—for me at least. In that sense what was gained was the algorithm.

I don’t really understand UMAP; for a good introduction to it see the original paper above and also this:

• Nikolay Oskolkov, How Exactly UMAP Works—and Why Exactly It Is Better Than tSNE, 3 October 2019.

tSNE is an older algorithm for taking clouds of data points in high dimensions and mapping them down to fewer dimensions so we can understand what’s going on. From the viewpoint of those working on genomics, the main good thing about UMAP is that it solves a bunch of problems that plagued tSNE. Oskolkov explains what these problems are and how UMAP deals with them. But he also alludes to the funny disconnect between these practicalities and the underlying theory:

My first impression when I heard about UMAP was that this was a completely novel and interesting dimension reduction technique which is based on solid mathematical principles and hence very different from tSNE which is a pure Machine Learning semi-empirical algorithm. My colleagues from Biology told me that the original UMAP paper was “too mathematical”, and looking at the Section 2 of the paper I was very happy to see strict and accurate mathematics finally coming to Life and Data Science. However, reading the UMAP docs and watching Leland McInnes talk at SciPy 2018, I got puzzled and felt like UMAP was another neighbor graph technique which is so similar to tSNE that I was struggling to understand how exactly UMAP is different from tSNE.

He then goes on and attempts to explain exactly why UMAP does so much better than tSNE. None of his explanation mentions category theory.

Since I don’t really understand UMAP or why it does better than tSNE, I can’t add anything to this discussion. In particular, I can’t say how much the category theory really helps. All I can do is explain a bit of the category theory. I’ll do that now, very briefly, just as a way to get a conversation going. I will try to avoid category-theoretic jargon as much as possible—not because I don’t like it or consider it unimportant, but because that jargon is precisely what’s stopping certain people from understanding Section 2.

I think it all starts with this paper by Spivak, which McInnes, Healy and Melville cite but for some reason don’t provide a link to:

• David Spivak, Metric realization of fuzzy simplicial sets.

Spivak showed how to turn a ‘fuzzy simplicial set’ into an ‘uber-metric space’ and vice versa. What are these things?

An ‘uber-metric space’ is very simple. It’s a slight generalization of a metric space that relaxes the usual definition in just two ways: it lets distances be infinite, and it lets distinct points have distance zero from each other. This sort of generalization can be very useful. I could talk about it a lot, but I won’t.

A fuzzy simplicial set is a generalization of a simplicial set.

A simplicial set starts out as a set of vertices (or 0-simplices), a set of edges (or 1-simplices), a set of triangles (or 2-simplices), a set of tetrahedra (or 3-simplices), and so on: in short, a set of n-simplices for each n. But there’s more to it. Most importantly, each n-simplex has a bunch of faces, which are lower-dimensional simplices.

I won’t give the whole definition. To a first approximation you can visualize a simplicial set as being like this:

But of course it doesn’t have to stop at dimension 3—and more subtly, you can have things like two different triangles that have exactly the same edges.

In a ‘fuzzy’ simplicial set, instead of a set of n-simplices for each n, we have a fuzzy set of them. But what’s a fuzzy set?

Fuzzy set theory is good for studying collections where membership is somewhat vaguely defined. Like a set, a fuzzy set has elements, but each element has a ‘degree of membership’ that is a number 0 < x ≤ 1. (If its degree of membership were zero, it wouldn't be an element!)

A map f: X → Y between fuzzy sets is an ordinary function, but obeying this condition: it can only send an element x ∈ X to an element f(x) ∈ Y whose degree of membership is greater than or equal to that of x. In other words, we don't want functions that send things to things with a lower degree of membership.

Why? Well, if I'm quite sure something is a dog, and every dog has a nose, then I'm must be at least equally sure that this dog has a nose! (If you disagree with this, then you can make up some other concept of fuzzy set. There are a number of such concepts, and I'm just describing one.)

So, a fuzzy simplicial set will have a set of n-simplices for each n, with each n-simplex having a degree of membership… but the degree of membership of its faces can't be less than its own degree of membership.

This is not the precise definition of fuzzy simplicial set, because I'm leaving out some distracting nuances. But you can get the precise definition by taking a nuts-and-bolts definition of simplicial set, like Definition 3.2 here:

• Greg Friedman, An elementary illustrated introduction to simplicial sets.

and replacing all the sets by fuzzy sets, and all the maps by maps between fuzzy sets.

If you like visualizing things, you can visualize a fuzzy simplicial set as an ordinary simplicial set, as in the picture above, but where an n-simplex is shaded darker if its degree of membership is higher. An n-simplex can’t be shaded darker than any of its faces.

How can you turn a fuzzy simplicial set into an uber-metric space? And how can you turn an uber-metric space into a fuzzy simplicial set?

Spivak focuses on the first question, because the answer is simpler, and it determines the answer to the second using some category theory. (Psst: adjoint functors!)

The answer to the first question goes like this. Say you have a fuzzy simplicial set. For each n-simplex whose degree of membership equals $a,$ you turn it into a copy of this uber-metric space:

$\{ (t_0, t_1, \dots, t_n) : t_0 + \cdots + t_n = - \log a , \; t_0, \ldots, t_n \geq 0 \} \subseteq \mathbb{R}^{n+1}$

This is really just an ordinary metric space: an n-simplex that’s a subspace of Euclidean (n+1)-dimensional space with its usual Euclidean distance function. Then you glue together all these uber-metric spaces, one for each simplex in your fuzzy simplical set, to get a big fat uber-metric space.

This process is called ‘realization’. The key here is that if an n-simplex has a high degree of membership, it gets ‘realized’ as a metric space shaped like a small n-simplex. I believe the basic intuition is that an n-simplex with a high degree of membership describes an (n+1)-tuple of things—its vertices—that are close to each other.

In theory, I should try to describe the reverse process that turns an uber-metric space into a fuzzy simplicial set. If I did, I believe we would see that whenever an (n+1)-tuple of things—that is, points of our uber-metric space—are close, they give an n-simplex with a high degree of membership.

If so, then both uber-metric spaces and fuzzy simplicial sets are just ways of talking about which collections of data points are close, and we can translate back and forth between these descriptions.

But I’d need to think about this a bit more to do a good job of going further, and reading the UMAP paper a bit more I’m beginning to suspect that’s not the main thing that practitioners need to understand. I’m beginning to think the most useful thing is to get a feeling for fuzzy simplicial sets! I hope I’ve helped a bit in that direction. They are very simple things. They are also closely connected to an idea from topological data analysis:

I should admit that McInnes, Healy and Melville tweak Spivak’s formalism a bit. They call Spivak’s uber-metric spaces ‘extended-pseudo-metric spaces’, but they focus on a special kind, which they call ‘finite’. Unfortunately, I can’t find where they define this term. They also only consider a special sort of fuzzy simplicial set, which they call ‘bounded’, but I can’t find the definition of this term either! Without knowing these definitions, I can’t comment on how these tweaks change things.

Applied Category Theory Meeting at UCR (Part 2)

30 September, 2019

Joe Moeller and I have finalized the schedule of our meeting on applied category theory:

Applied Category Theory, special session of the Fall Western Sectional Meeting of the AMS, U. C. Riverside, Riverside, California, 9–10 November 2019.

It’s going to be really cool, with talks on everything from brakes to bicategories, from quantum physics to social networks, and more—with the power of category theory as the unifying theme!

You can get information on registration, hotels and such here. If you’re coming, you might also want to attend Eugenia Cheng‘s talk on the afternoon of Friday November 8th.   I’ll announce the precise title and time of her talk, and also the location of all the following talks, as soon as I know!

In what follows, the person actually giving the talk has an asterisk by their name. You can click on talk titles to see abstracts of the talks.

Saturday November 9, 2019, 8:00 a.m.-10:50 a.m.

• 8:00 a.m.David I. Spivak*, Massachusetts Institute of Technology
• 9:00 a.m.Brendan Fong*, Massachusetts Institute of Technology
David I. Spivak, Massachusetts Institute of Technology
• 9:30 a.m.Gabriel C. Drummond-Cole, IBS Center for Geometry and Physics
Philip Hackney*, Department of Mathematics, University of Louisiana at Lafayette
• 10:00 a.m.
Duality of relations.
Alexander Kurz*, Chapman University
• 10:30 a.m.Tobias Fritz*, Perimeter Institute for Theoretical Physics

Saturday November 9, 2019, 3:00 p.m.-5:50 p.m.

Sunday November 10, 2019, 8:00 a.m.-10:50 a.m.

Sunday November 10, 2019, 2:00 p.m.-4:50 p.m.

Applied Category Theory 2019 Program

3 July, 2019

Bob Coecke, David Spivak, Christina Vasilakopoulou and I are running a conference on applied category theory:

Applied Category Theory 2019, 15–19 July, 2019, Lecture Theatre B of the Department of Computer Science, 10 Keble Road, Oxford.

You can now see the program here, or below. Hope to see you soon!

Enriched Lawvere Theories

16 May, 2019

My grad student Christian Williams and I finished this paper just in time for him to talk about it at SYCO:

• John Baez and Christian Williams, Enriched Lawvere theories for operational semantics.

Abstract. Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph-enriched Lawvere theory describes structures that have a graph of operations of each arity, where the vertices are operations and the edges are rewrites between operations. Enriched theories can be used to equip systems with operational semantics, and maps between enriching categories can serve to translate between different forms of operational and denotational semantics. The Grothendieck construction lets us study all models of all enriched theories in all contexts in a single category. We illustrate these ideas with the SKI-combinator calculus, a variable-free version of the lambda calculus, and with Milner’s calculus of communicating processes.

When Mike Stay came to U.C. Riverside to work with me about ten years ago, he knew about computation and I knew about category theory, and we started trying to talk to each other. I’d heard that categories and computer science were deeply connected: for example, people like to say that the lambda-calculus is all about cartesian closed categories. But we soon realized something funny was going on here.

Computer science is deeply concerned with processes of computation, and category theory uses morphisms to describe processes… but when cartesian closed categories are applied to the lambda calculus, their morphisms do not describe processes of computation. In fact, the process of computation is effectively ignored!

We decided that to fix this we could use 2-categories where

• objects are types. For example, there could be a type of integers, INT. There could be a type of pairs of integers, INT × INT. There could also be a boring type 1, which represents something there’s just one of.

• morphisms are terms. For example, a morphism f: 1 → INT picks out a specific natural number, like 2 or 3. There could also be a morphism +: INT × INT → INT, called ‘addition’. Combining these, we can get expressions like 2+3.

• 2-morphism are rewrites. For example, there could be a rewrite going from 2+3 to 5.

Later Mike realized that instead of 2-categories, it can be good to use graph-enriched categories: that is, things like categories where instead of a set of morphisms from one object to another, we have a graph.

In other words: instead of hom-sets, a graph-enriched category has ‘hom-graphs’. The objects of a graph-enriched category can represent types, the vertices of the hom-graphs can represent terms, and the edges of the hom-graphs can represent rewrites.

Mike teamed up with Greg Meredith to write a paper on this:

• Mike Stay and Greg Meredith, Representing operational semantics
with enriched Lawvere theories
.

Christian decided to write a paper building on this, and I’ve been helping him out because it’s satisfying to see an old dream finally realized—in a much more detailed, beautiful way than I ever imagined!

The key was to sharpen the issue by considering enriched Lawvere theories. Lawvere theories are an excellent formalism for describing algebraic structures obeying equational laws, but they do not specify how to compute in such a structure, for example taking a complex expression and simplifying it using rewrite rules. Enriched Lawvere theories let us study the process of rewriting.

Maybe I should back up a bit. A Lawvere theory is a category with finite products $T$ generated by a single object $t$, for ‘type’. Morphisms $t^n \to t$ represent n-ary operations, and commutative diagrams specify equations these operations obey. There is a theory for groups, a theory for rings, and so on. We can specify algebraic structures of a given kind in some ‘context’—that is, in some category $C$ with finite products—by a product-preserving functor $\mu: T \to C.$ For example, if $T$ is the theory of groups and $C$ is the category of sets then such a functor describes a group, but if $C$ is the category of topological space then such a functor describes a topological group.

All this is a simple and elegant form of what computer scientists call denotational semantics: roughly, the study of types and terms, and what they signify. However, Lawvere theories know nothing of operational semantics: that is, how we actually compute. The objects of our Lawvere are types and the morphisms are terms. But there are no rewrites going between terms, only equations!

This is where enriched Lawvere theories come in. Suppose we fix a cartesian closed category V, such as the category of sets, or the category of graphs, or the category of posets, or even the category of categories. Then V-enriched category is a thing like a category, but instead of having a set of morphisms from any object to any other object, it has an object of V. That is, instead of hom-sets it can have hom-graphs, or hom-posets, or hom-categories. If it has hom-categories, then it’s a 2-category—so this setup includes my original dream, but much more!

Our paper explains how to generalize Lawvere theories to this enriched setting, and how to use these enriched Lawvere theories in operational semantics. We rely heavily on previous work, especially by Rory Lucyshyn-Wright, who in turn built on work by John Power and others. But we’re hoping that our paper, which is a bit less high-powered, will be easier for people who are familiar with category theory but not yet enriched categories. The novelty lies less in the math than its applications. Give it a try!

Here is a small piece of a hom-graph in the graph-enriched theory of the SKI combinator calculus, a variable-free version of the lambda calculus invented by Moses Schönfinkel and Haskell Curry back in the 1920s:

Symposium on Compositional Structures 4: Program

11 May, 2019

Here’s the program for this conference:

Symposium on Compositional Structures 4, 22–23 May, 2019, Chapman University, California. Organized by Alexander Kurz.

A lot of my students and collaborators are speaking here! The meeting will take place in Beckman Hall 101.

Wednesday May 22, 2019

• 10:30–11:30 — Registration.

• 11:30–12:30 — John Baez, “Props in Network Theory“.

• 12:30–1:00 — Jade Master, “Generalized Petri Nets”.

• 1:00–2:00 — Lunch.

• 2:00–2:30 — Christian Williams, “Enriched Lawvere Theories for Operational Semantics”.

• 2:30–3:00 — Kenny Courser, “Structured Cospans”.

• 3:00–3:30 — Daniel Cicala, “Rewriting Structured Cospans”.

• 3:30–4:00 — Break.

• 4:00–4:30 — Samuel Balco and Alexander Kurz, “Nominal String Diagrams”.

• 4:30–5:00 — Jeffrey Morton, “2-Group Actions and Double Categories”.

• 5:00–5:30 — Michael Shulman, “All (∞,1)-Toposes Have Strict Univalent Universes”.

• 5:30–6:30 — Reception.

Thursday May 23, 2019

• 9:30–10:30 — Nina Otter, “A Unified Framework for Equivalences in Social Networks”.

• 10:30–11:00 — Kohei Kishida, Soroush Rafiee Rad, Joshua Sack and Shengyang Zhong, “Categorical Equivalence between Orthocomplemented Quantales and Complete Orthomodular Lattices”.

• 11:00–11:30 — Break.

• 11:30–12:00 — Cole Comfort, “Circuit Relations for Real Stabilizers: Towards TOF+H”.

• 12:00–12:30 — Owen Biesel, “Duality for Algebras of the Connected Planar Wiring Diagrams Operad”.

• 12:30–1:00 — Joe Moeller and Christina Vasilakopoulou, “Monoidal Grothendieck Construction”.

• 1:00–2:00 — Lunch.

• 2:00–3:00 — Tobias Fritz, “Categorical Probability: Results and Challenges”.

• 3:00–3:30 — Harsh Beohar and Sebastian Küpper, “Bisimulation Maps in Presheaf Categories”.

• 3:30–4:00 — Break.

• 4:00–4:30 — Benjamin MacAdam, Jonathan Gallagher and Rory Lucyshyn-Wright, “Scalars in Tangent Categories”.

• 4:30–5:00 — Jonathan Gallagher, Benjamin MacAdam and Geoff Cruttwell, “Towards Formalizing and Extending Differential Programming via Tangent Categories”.

• 5:00–5:30 — David Sprunger and Shin-Ya Katsumata, “Differential Categories, Recurrent Neural Networks, and Machine Learning”.