Thirteenth Symposium on Compositional Structures (SYCO 13)

London, UK
24-25 April, 2025

The Symposium on Compositional Structures is a series of interdisciplinary meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. You could submit work-in-progress, or a recently completed paper, or even a PhD or Masters thesis.

While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in particular from the perspective of category theory:

This new series aims to bring together the communities behind many previous successful events which have taken place over the last decade, including Categories, Logic and Physics, Categories, Logic and Physics (Scotland), Higher-Dimensional Rewriting and Applications, String Diagrams in Computation, Logic and Physics, Applied Category Theory, Simons Workshop on Compositionality, Yorkshire and Midlands Category Theory Seminar, and the Peripatetic Seminar in Sheaves and Logic.

This event follows SYCO 1 in Birmingham, SYCO 2 in Strathclyde, SYCO 3 in Oxford, SYCO 4 in California, SYCO 5 in Birmingham, SYCO 6 in Leicester, SYCO 8 in Tallinn, SYCO 9 in Como, SYCO 10 in Edinburgh, SYCO 11 in Palaiseau, and SYCO 12 in Birmingham.

Invited speakers

Elena Di Lavore Martti Karvonen
Elena Di Lavore Martti Karvonen
University of Oxford University College London
Partial Markov categories What’s going on with categorical composable cryptography?

Important dates

All deadlines are 23:59 anywhere-on-earth on the given dates.

Registration

In-person registration is closed, but you can still register to attend online by filling in the following form: SYCO 13 Registration Form

Accepted talks

Schedule

All times are in British Summer Time (UTC+1).

Thursday 24 April

10:30-11:00
REGISTRATION (and coffee)
Chair: Alexandra Silva
11:00-12:00
Elena Di Lavore
Partial Markov categories
[Slides]
Partial Markov categories are a synthetic theory of stochastic processes with constraints, observations and updates.

They unify the perspective of Markov categories on stochastic processes with that of cartesian restriction categories on partial processes. I will give an overview of these structures and show how they relate to each other. The structure of partial Markov categories allows bayesian reasoning: it gives a synthetic Bayes theorem.
12:00-12:30
Mika Bohinen and Paolo Perrone
Categorical algebra of conditional probability
[Slides]
In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and statistics. In this paper, we connect some ideas from categorical algebra, namely weakly cartesian functors and natural transformations, to the idea of conditioning in probability theory, using Markov categories and probability monads. First of all, we show that under some conditions, the monad associated to a Markov category with conditionals has a weakly cartesian functor and weakly cartesian multiplication (a condition known as Beck-Chevalley, or BC). In particular, we show that this is the case for the Giry monad on standard Borel spaces. We then connect this theory to existing results on statistical experiments. We show that for deterministic statistical experiments, the so-called standard measure construction (which can be seen as a generalization of the "hyper-normalizations" introduced by Jacobs) satisfies a universal property, allowing an equivalent definition which does not rely on the existence of conditionals.
12:30-13:30
LUNCH
Chair: Amar Hadzihasanovic
13:30-14:00
Razin Shaikh, Lia Yeh and Stefano Gogioso
The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation
[Slides]
While the ZX and ZW calculi have been effective as graphical reasoning tools for finite-dimensional quantum computation, the possibilities for continuous-variable quantum computation (CVQC) in infinite-dimensional Hilbert space are only beginning to be explored. In this work, we formulate a graphical language for CVQC. Each diagram is an undirected graph made of two types of spiders: the Z spider from the ZX calculus defined on the reals, and the newly introduced Fock spider defined on the natural numbers. The Z and X spiders represent functions in position and momentum space respectively, while the Fock spider represents functions in the discrete Fock basis. In addition to the Fourier transform between Z and X, and the Hermite transform between Z and Fock, we present exciting new graphical rules capturing heftier CVQC interactions. We ensure this calculus is complete for all of Gaussian CVQC interpreted in infinite-dimensional Hilbert space, by translating the completeness in affine Lagrangian relations by Booth, Carette, and Comfort. Applying our calculus for quantum error correction, we derive graphical representations of the Gottesman-Kitaev-Preskill (GKP) code encoder, syndrome measurement, and magic state distillation of Hadamard eigenstates. Finally, we elucidate Gaussian boson sampling by providing a fully graphical proof that its circuit samples submatrix hafnians.
14:00-14:30
Thea Li
The Category of Finite Dimensional Operator Spaces
[Slides]
We investigate the semantical properties of the category of finite dimensional operator spaces, in particular, we prove that it is a non-degenerate model of multiplicative additive linear logic. We also argue that this category gives a good basis for modeling BV-logic, an extension of multiplicative linear logic, by showing that it is a BV-category.
14:30-15:00
Muhammad Hamza Waseem, Caterina Puca, Lia Yeh, Selma Dündar-Coecke, Bob Coecke, Aleks Kissinger, Stefano Gogioso, Sieglinde Pfaendler and Thomas Cervoni
Teaching Quantum Theory with a Compositional Lens: Experimental Evidence Supporting the Effectiveness of Quantum Picturalism
[Slides]
This talk presents a pedagogical experiment that uses Quantum Picturalism (QP)—a compositional formalism of quantum theory—to test whether quantum physics and technologies can be effectively taught to high school students. Conducted in 2023, this experiment involved students aged 16-18. 54 randomly selected students from the UK attended two-hour online classes and tutorials weekly for eight weeks, and were then assessed using Oxford postgraduate quantum physics exam questions. The results showed that over 80% of the students passed, with about half earning a distinction.

The talk will provide an overview of our pedagogical experiment, the subsequent developments in the education and policy sectors, and the future prospects of the QP education programme. It will also include examples of QP educational material, highlighting how quantum ideas are expressed and communicated through a compositional lens. We hope this talk will showcase the merits and potential of the compositional turn in quantum physics, especially in pedagogy.
15:00-15:30
BREAK
Chair: Nick Hu
15:30-16:00
Max Demirdilek and Christoph Schweigert
Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories
[Slides]
Grothendieck-Verdier categories (also known as star-autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules over vertex operator algebras.

In this paper, we develop a surface-diagrammatic calculus for Grothendieck-Verdier categories, extending the string-diagrammatic calculus of Joyal and Street for rigid monoidal categories into a third dimension. This extension naturally arises from the non-invertibility of coherence data in Grothendieck-Verdier categories.

We show that key properties of Frobenius algebras in rigid monoidal categories carry over to the Grothendieck-Verdier setting. Moreover, we introduce higher Frobenius-Schur indicators for suitably finite k-linear pivotal Grothendieck-Verdier categories and prove their invariance under pivotal Frobenius linearly distributive equivalences.

The proofs are carried out using the surface-diagrammatic calculus.
16:00-16:30
Clémence Chanavat and Amar Hadzihasanovic
Diagrammatic (∞,n)-categories
[Slides]
I will give an overview of the diagrammatic model of (∞,n)-categories. This is designed to share (or improve) the good features of strict higher categories—such as a strong pasting theorem enabling explicit diagrammatic reasoning, and an expressive language for cellular models—while provably satisfying the homotopy hypothesis, and hopes to function as a bridge between non-algebraic and algebraic models.
16:30-17:00
Thibaut Benjamin, Ioannis Markakis, Wilfred Offord, Chiara Sarti and Jamie Vicary
Naturality for higher-dimensional path types
[Slides]
We define a naturality construction for the operations of weak omega-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product with a directed interval, and behaves logically as a globular analogue of Reynolds parametricity. Our construction operates as a "power tool" to support construction of terms with geometrical structure, and we use it to define composition operations for cylinders and cones in omega-categories. The machinery can generate terms of high complexity, and we have implemented our construction in a proof assistant, which verifies that the generated terms have the correct type. All our results can be exported to homotopy type theory, allowing the explicit computation of complex path type inhabitants.
18:30
EVENING SOCIAL

Friday 25 April

Chair: Mina Abbaszadeh
09:30-10:30
Martti Karvonen
What’s going on with categorical composable cryptography?
[Slides]
In this talk we discuss prior and ongoing work on categorical cryptography. We first recap the categorical framework formalizing the simulation paradigm of cryptography. An instructive example is given by the one-time pad, whose composable security follows from the axioms of an Hopf algebra with an integral in a symmetric monoidal category. We then survey ongoing work, including (i) the search for further pictorial proofs in cryptography, based on notions defined in terms of computational indistinguishability such as pseudorandomness and (ii) capturing (variants) of other approaches to composable security under the categorical framework.
10:30-11:00
Fatimah Rita Ahmadi
Typing Tensor Calculus in 2-Categories
[Slides]
To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallelized computations, we adopt an approach that treats elements of linear algebra, such as matrices, as morphisms in the category of matrices, Mat_k. This framework is further extended by generalizing the results to arbitrary monoidal semiadditive categories. To enrich this perspective and accommodate higher-rank matrices (tensors), we define semiadditive 2-categories, where matrices T_ij are represented as 1-morphisms, and tensors with four indices T_ijkl as 2-morphisms. This formalization provides an index-free, typed linear algebra framework that includes matrices and tensors with up to four indices. Furthermore, we extend the framework to monoidal semiadditive 2-categories and demonstrate detailed operations and vectorization within the 2-category of 2Vec introduced by Kapranov and Voevodsky.

If time permits, I will discuss generalisation of this work to cartesian closed bicategories.
11:00-11:30
BREAK
Chair: Mehrnoosh Sadrzadeh
11:30-12:00
Tiffany Duneau
A Compositional Approach to Reading Comprehension Tasks Using the DisCoCirc Natural Language Processing Framework
[Slides]
The DisCoCirc framework for natural language processing allows the construction of compositional models of text, by combining units for individual words together according to the grammatical structure of the text to be read by the model. The compositional nature of a model can give rise to two things: compositional generalisation - the ability of a model to generalise outside its training distribution by learning compositional rules underpinning the entire data distribution - and compositional understandability - making sense of how the model works by inspecting its modular components in isolation and the processes through which they are combined.

We consider how well DisCoCirc models can learn different types of compositional behaviour. We compare both quantum circuit based models, as well as classical neural networks under a series of training paradigms, on a dataset derived from one of the bAbI tasks, extended to test a series of aspects of compositionality.

Both architectures score within 5% of one another on the productivity and substitutivity tasks, but differ for the systematicity and overgeneralisation tasks. Overall, we find the neural models are more prone to overfitting the Train data.

Finally, we inspect the trained models, both by comparing them to manually-constructed perfect compositional models, and by considering how the model components interact with one another, explaining how the models behave and exhibiting the models' compositional understandability.
12:00-12:30
Jake Araujo-Simon
Compositional Nonlinear (Audio) Signal Processing with Volterra Series
[Slides]
We present a compositional theory of nonlinear audio signal processing based on a categorification of the Volterra series. We begin by augmenting the classical definition of the Volterra series so that it is functorial with respect to a base category whose objects are temperate distributions and whose morphisms are certain linear transformations. This motivates the derivation of formulae describing how the outcomes of nonlinear transformations are affected if their input signals are linearly processed – e.g., translated, modulated, sampled, or periodized. We then consider how nonlinear audio systems, themselves, change, and introduce as a model thereof the notion of morphism of Volterra series, which we exhibit as both a type of lens map and natural transformation. We show how morphisms can be parameterized and used to generate indexed families (e.g., sequences) of Volterra series, which are well-suited to model nonstationary or time-varying nonlinear phenomena. We then describe how Volterra series and their morphisms organize into a category, which we call Volt. We exhibit the operations of sum, product, and series composition of Volterra series as monoidal products on Volt, and identify, for each in turn, its corresponding universal property. In particular, we show that the series composition of Volterra series is associative. We then bridge between our framework and the subject at the heart of audio signal processing: time-frequency analysis. Specifically, we show that a known equivalence, between a class of second-order Volterra series and the bilinear time-frequency distributions (TFDs), can be extended to one between certain higher-order Volterra series and the so-called polynomial TFDs. We end by outlining potential avenues for future work, including the incorporation of nonlinear system identification techniques and the potential extension of our theory to the settings of graph and topological audio signal processing.
12:30-13:00
Morgan Rogers
Building models from finite pieces
[Slides]
In a previous iteration of SYCO, I explained the relationship between classifying toposes, categories of monoid actions and a classical result from model theory: that there is a Galois connection between subgroups of the group of automorphisms of a model (of a first-order theory) and "extensions" of that model.

In this talk I would like to present a topos-theoretic perspective on another piece of model theory, namely Fraissé theory, which provides sufficient conditions for the existence of a special "ultrahomogeneous" model of a theory to exist. This is a model that contains all finite (or finitely generated) models as submodels in a symmetric way, such as the Rado graph.

Usually (in work of Caramello or Kubis, say) the category-theoretic presentation of Fraisse theory again involves automorphism groups, so the main novelty of my presentation will be to show what extra flexibility moving from groups to monoids affords us in the Fraisse construction. This is based on Chapter 6 of my thesis*, which as yet has not been reworked into a published form (so is technically work in progress).

*https://irinsubria.uninsubria.it/retrieve/e2188be8-0f66-4564-e053-6605fe0a49d6/Thesis_caricamento.pdf
13:00-14:00
LUNCH
Chair: Jamie Vicary
14:00-14:30
Chris Purdy and Stefania Damato
Distributive Laws of Monadic Containers
[Slides]
Containers are used to carve out a class of strictly positive data types in terms of shapes and positions. They can be interpreted via a fully-faithful functor into endofunctors on Set. Monadic containers are an extension of containers whose interpretation as a Set functor carries a monad structure. The category of containers is closed under container composition and is a monoidal category, whereas monadic containers do not in general compose.

In this paper, we develop a characterisation of distributive laws of monadic containers. Distributive laws were introduced as a sufficient condition for the composition of the underlying functors of two monads to also carry a monad structure. Our development parallels Ahman and Uustalu's characterisation of distributive laws of directed containers, i.e. containers whose Set functor interpretation carries a comonad structure. Furthermore, by combining our work with theirs, we construct characterisations of mixed distributive laws (i.e. of directed containers over monadic containers and vice versa), thereby completing the `zoo' of distributive laws from a container perspective.

We have found these characterisations amenable to development of existence and uniqueness proofs of distributive laws, particularly in the mechanised setting of Cubical Agda, in which most of the theory of this paper has been formalised.
14:30-15:00
Filippo Bonchi, Cipriano Junior Cioffo, Alessandro Di Giorgio, Elena Di Lavore
Tape Diagrams for Monoidal Monads
[Slides]
Tape diagrams provide a graphical representation for arrows of rig categories, namely categories equipped with two monoidal structures, $\oplus$ and $\otimes$, where $\otimes$ distributes over $\oplus$. However, their applicability is limited to categories where $\oplus$ is a biproduct, i.e., both a categorical product and a coproduct. In this work, we extend tape diagrams to deal with Kleisli categories of symmetric monoidal monads, presented by algebraic theories.
15:00-15:30
Nick Hu, Alex Rice, Calin Tataru and Dan Ghica
sd-visualiser: interactive hypergraph visualisation for programs as string diagrams
We present sd-visualiser, a tool for transforming programs as syntax into hierarchical hypergraphs, and rendering the output as interactive string diagrams. In doing so, we define a toy language sd-lang for the description of such hypergraphs as programs.

Many compilers use graph-based intermediate representations (IRs) of programs internally for the purposes of code generation, program analysis, and optimisation. We show how our methods extend to real-world IRs of programs generated by the LLVM/MLIR compiler toolchain to visualise programs with varying combinations of compiler optimisations applied, providing a tool to explore these graph-based IRs which is formally grounded in categorical semantics.

We implement the correspondence between our notion of hierarchical hypergraph and terms in a traced cartesian closed category, whereby connectivity of hyperedges is translated into a foliation of monoidal terms with explicit copying, deletion, and swapping. Our approach applies computational methods for drawing string diagrams, which is more principled than existing ad-hoc GraphViz-based approaches, allowing us to effectively combine dataflow and control flow with sharing in one aesthetically pleasing picture. Moreover, the interactive nature of our tool allows for collapsing/expanding hierarchical nodes, and iterative refinement for focusing on a particular region of interest within a large program delineated by hypergraph connectivity.

For more information, please see https://github.com/sd-visualiser/sd-visualiser.
A web version of the tool can be accessed at: https://sd-visualiser.github.io/sd-visualiser/
15:30
END

Local information

Talks will take place on the main campus of University College London, in Lecture theatre G01, located at 66-72 Gower St, London WC1E 6EA.

The event is catered. Coffee, tea and snacks will be offered during coffee breaks. At lunch, there will be a choice of sandwiches and cold dishes catering to different dietary requirements.

Transport

Tube

The closest tube stations to UCL sites are Goodge St (Northern Line), Russell Square (Piccadilly line), Euston (Northern and Victoria lines), Euston Square (Hammersmith and City, Metropolitan and Circle lines), and Warren Street (Northern and Victoria lines). More information and a journey planner can be found at www.tfl.gov.uk/tube.

You can also find directions from these stations to UCL locations: Russell Square Station, Euston Station, Euston Square Station, Kings Cross Station, Warren Street Station.

By Bus

UCL is served by many Transport for London bus routes. There are several lines (e.g. 68, 91) travelling North-South with stops near Russell Square, and bus lines (e.g. 18, 30) travelling East-West with stops at Euston station. More information and a journey planner can be found at www.tfl.gov.uk/buses.

By National and International Rail

London has many mainline rail stations. Most of these are a short journey away from UCL, with the stations at Euston, King’s Cross and St Pancras being within easy walking distance (10-15min). See www.nationalrail.co.uk for more information.

For travel outside of the UK, there are trains from St Pancras to France (Paris, Lille), Belgium (Brussels), and the Netherlands (Amsterdam, Rotterdam). See wwww.eurostar.com.

By Air

Below are directions to UCL from London’s five main airports. There are bus and taxi alternatives from each of the airports. Details of trains from airports to central London can also be obtained from National Rail at www.nationalrail.co.uk.

Accommodation

There are plenty of hotels in Bloomsbury, near UCL's main campus. Some mid-range suggestions below:

Alternatively, there should be several options available on Airbnb. If you choose not to stay near UCL, we recommend making sure your accommodation has good transport links to the venue. London is a large city, and even journeys between central areas can easily take over 40 minutes.

Social activities

Pub social

If you are in London on the evening before the workshop, Wednesday 23 April, please join us from 17:30 at The Marquis Cornwallis, a pub in Bloomsbury, near UCL.

Evening meal

On the evening of Thursday 24 April from 18:30, we will have dinner at Sagar West End, and Indian vegetarian restaurant, near the venue. Note that participants are expected to pay for themselves.

Attendees

Attendees marked with an asterisk * will attend physically.

Programme committee

Steering committee

The symposium is managed by the following people. If you have a general question about SYCO, or if you want to propose to host a future iteration, please get in touch with a member of the steering committee.