The **Edinburgh category theory seminar** runs on some Wednesdays from 12:00 to 13:00 in the Bayes
Centre, Room 5.46. Most talks will also be livestreamed via Zoom. Please note that we won't be meeting on strike
days.

The seminar is intended to be an informal environment where people interested in category theory can share ideas. Most of our speakers are internal, with some occasional guests. Talks span a wide range of topics, from derived algebraic geometry to theoretical computer science, so they should be accessible to anyone with some category theory basics. There is no requirement to talk about cutting-edge research, and, in particular, PhD students should feel comfortable giving an introductory talk on some standard categorical topic.

This is one of several seminars run within the Hodge Institute at Edinburgh. The details of these can be found on the Hodge seminars website.

For more details or to propose a talk, please email Adrián or Tom. The old website for the Edinburgh category theory seminar can be found here.

### Winter/Spring 2024

Dynamicalization is the process of taking a concept that applies to static objects (such as spaces) and extending it to objects that evolve through time (such as spaces equipped with an action by \(\mathbb{Z}\) or \(\mathbb{R}\)). The term was introduced, I believe, by Masaki Tsukamoto.

Dynamicalization is similar in flavour to categorification, and should be amenable to a systematic categorical approach. It could even have the same massive scope as categorification – even though it has received only a tiny fraction of the attention. In fact, there is a sense in which dynamicalization appears to be dual to categorification. I will explain what I understand so far, give some examples, and try to convey what I think is the promise of this big idea. There will be many more questions than answers.

What does it mean for a mathematical function or operation to be 'computable'? The received wisdom is that all reasonable models of computation embody essentially the same notion of computability (the Church-Turing one). However, there are contexts in which this simple view is no longer adequate: e.g. the setting of higher-order functions, where different (natural) models may embody genuinely different concepts of 'computability'. Not only are many of these models naturally viewed as categories, but it is also to think in terms of a big 2-category of such models, in order to understand how the different notions of computability relate to one another.

I will give a high-level overview of this whole programme of research, which has occupied me for several decades now. I will focus mainly on the mathematics, but will also touch on the motivations e.g. from the study of programming languages.

This talk bridges between two major paradigms in computation, the functional, at basis computation from input to output, and the interactive, where computation reacts to its environment while underway. Central to any compositional theory of interaction is the dichotomy between a system and its environment. Concurrent games and strategies address the dichotomy in fine detail, very locally, in a distributed fashion, through distinctions between Player moves (events of the system) and Opponent moves (those of the environment). A functional approach has to handle the dichotomy much more ingeniously, through its blunter distinction between input and output. This has led to a variety of functional approaches, specialised to particular interactive demands. Through concurrent games we can more clearly see what separates and connects the differing paradigms, and show how:

- to lift functions to strategies; this helps in describing and programming strategies by functional techniques.
- several paradigms of functional programming and logic arise naturally as subcategories of concurrent games, including stable domain theory; nondeterministic dataflow; geometry of interaction; the dialectica interpretation; lenses and optics; and their extensions to containers in dependent lenses and optics.
- to transfer enrichments of strategies (such as to probabilistic, quantum or real-number computation) to functional cases.

The talk will focus on the second and third points above. (Details can be found in the expanded version of my LICS'23 paper at arXiv:2202.13910)

Monads are generalisations of algebraic theories described by some operations on a set and some axioms on those operations, such as the theory of groups. One way to say this is that the forgetful functor from \(\mathbf{Grp}\) to \(\mathbf{Set}\) is monadic. If the right Kan extension of a functor \(G\) along itself exists, then it has a canonical monad structure: the codensity monad of \(G\). This gives a universal replacement of \(G\) by a monadic functor. I will introduce a related construction which will allow us to pushforward a monad along a functor, and explain some of its properties.

This process produces some surprising results, the most well-known being that the monadic-over-\(\mathbf{Set}\) replacement of the category of finite sets is the category of compact Hausdorff spaces. I will review this, and explore how pushing forward monads on finite sets to Set gives their topological analogues. Time permitting, I will present the more algebraically flavoured example of the codensity monad of the inclusion of \(\mathbf{Field}\) into \(\mathbf{Ring}\).

I will explain the motivation and main ideas behind recent joint work with Chris Heunen (arXiv:2401.06584) that characterises the category of finite-dimensional Hilbert spaces and linear contractions. The axioms are about simple category-theoretic structures and properties. In particular, they do not refer to norms, continuity, dimension, or real numbers. The proof is noteworthy for the new way that the scalars are identified as the real or complex numbers. Instead of resorting to Solèr’s theorem, which is an opaque result underpinning similar characterisations of other categories of Hilbert spaces, suprema of bounded increasing sequences of scalars are explicitly constructed using directed colimits of contractions. To keep the talk accessible, I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed.

The algebra of functions on any group is a prototypical example of a commutative Hopf algebra, while a (in general non-commutative) Hopf algebra can be interpreted as a deformation of a group. In this talk, I will review Hopf algebras and describe nerves of Hopf algebras, which are braided lax monoidal functors from an extension of the simplex category. Finally, if time permits, I will give an application of this formalism to deformation quantization of Lie bialgebras/Poisson Lie groups. Based on a joint work with Severa.

Operads are combinatorial gadgets that control algebraic theories. They were first introduced by Boardman–Vogt and May to classify homotopy types of iterated loop spaces and have since become an invaluable tool for doing categorical algebra. Their \(\infty\)-categorical incarnation, \(\infty\)-operads, subsumes the classical notion and has been a major player in the recent renaissance of homotopy-coherent algebra.

The theory of \(\infty\)-operads is controlled by the category of pointed finite sets. Naturally, one may ask which categories give rise to an "operad-like" theory. As an answer to this question, I will describe Barwick's notion of an operator category \(\Phi\) and its associated theory of \(\Phi\)-operads. This is a natural generalization of \(\infty\)-operads, subsuming many known variants of the notion, for example non-symmetric operads. I will not assume knowledge of \(\infty\)-categories for this talk and will explain everything as I go.

In this talk, my aim is to introduce the notion of a braided module category. To illustrate, I will describe an obstacle I faced in finishing my thesis and how braided module categories helped me to overcome this, capturing an essential symmetry of the problem. Along the way we will meet a higher category which describes the notion of Morita equivalence for (symmetric, braided, etc) monoidal categories.

Quantum computations have two ingredients: unitary gates forming reversible circuits, and irreversible measurements. The theory of quantum computation is at heart therefore about how these ingredients combine. Rig categories track these combinatorics. Measurements are explained by a universal construction on rig categories that 'hides' parts of objects. For circuits, free rig categories satisfying a small number of simple equations are universal, sound and complete for various gate sets. The precision with which a circuit approximates a unitary matrix is linked to ring extensions by a universal construction on rig categories that 'adds ancillas'. (Based on joint papers and ongoing work with Pablo Andres-Martinez, Jacques Carette, Robin Kaarsgaard, Neil Julien Ross, and Amr Sabry.)

This will be a expository talk; there will be few, if any, new results. I will describe some incarnations of

### Autumn 2023

This is a speculative talk inspired by recent work of Arij Benkhadra and Isar Stubbe. On the one hand, the Banach fixed point theorem is a staple of undergraduate courses on metric spaces. On the other, metric spaces can usefully be seen as enriched categories. This raises the question of whether the Banach fixed point theorem can be understood categorically. I will talk about the search for an answer, opening up avenues of investigation without, yet, being able to say what lies at the end of them.

The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A "rig" is a "ring without negatives", and the free rig on one generator is \(\mathbb{N}[x]\), the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of "symmetric 2-rig", and it turns out that the category of Schur functors is the free symmetric 2-rig on one generator. Thus, in a certain sense, Schur functors are the next step after polynomials.

This will be an informal talk about research in progress. For the last few years, I've been obsessed with the idea that it is possible to make sense of quantum field theories in very general geometric settings. I'm getting closer to understanding the algebra of observables of such theories, and I've been amused to identify some simple categorical structures that I hadn't seen before. But maybe some of you have!

Contractions play an important role in functional analysis. In this talk, I will introduce an abstract notion of contraction in a dagger category with finite dagger products. The talk will be accessible to everyone in attendance, not assuming prior familiarity with dagger categories, but instead introducing the relevant concepts with reference to the category of Hilbert spaces and bounded linear maps. I will motivate and report on ongoing work connecting directed colimits in the wide subcategories of dagger monomorphisms and of contractions.

I will review basic concepts of monad theory, focusing on comparison theorem. Then I will explain how this tool from abstract category theory can be used for more practical purposes and deduce some natural constructions for abelian and triangulated categories. Specifically, I plan to discuss

- relation between equivariant and derived categories,
- cohomological descent for a morphism of schemes, and
- scalar extension for linear categories.

I hope that the talk will be as elementary as possible. Besides of folklore knowledge, it is based on papers “Cohomological descent theory for a morphism of stacks and for equivariant derived categories”, arXiv:1103.3135 and “On equivariant triangulated categories”, arXiv:1403.7027.

Filtered colimits show up in most areas of mathematics. The root of their importance comes from the fact that, in \(\mathbf{Set}\), finite limits commute with filtered colimits. I will begin by reviewing this result. I will then give an introduction to Lawvere theories: categorical gadgets able to encode algebraic theories, such as the theory of groups. Every Lawvere theory gives rise to a monad on \(\mathbf{Set}\), whose algebras are the models for the theory in \(\mathbf{Set}\). I will explain why the monads that arise in this way are precisely the finitary monads, i.e. those whose underlying functor preserves filtered colimits, and why, in their categories of algebras, finite limits again commute with filtered colimits.

Order and topology both abound in mathematics, often even together. Combined, they form the notion of an ordered topological space. How can this notion be suitably generalised to the theory of point-free topology? In this talk we will discuss one such possibility. To keep the talk accessible, we start with an introduction to the theory of locales, focusing on their adjunction with topological spaces. After that, we show how this adjunction can be extended to certain categories of ordered spaces and ordered locales. To finish the talk, time permitting, we highlight some ongoing work building on this framework. (Based on joint work with Chris Heunen.)

A Bridgeland stability condition on a triangulated category D gives you a way to build your category out of simpler “stable” objects. Different choices of which objects are your building blocks give rise to different stability conditions. It turns out that the space of all stability conditions on D, Stab(D), has the structure of a complex manifold, giving us a way to extract geometry from our category. In this talk, I will introduce stability conditions, and explain a way to relate actions of finite abelian groups on D to the geometry of Stab(D). Time permitting, I will discuss joint work with Edmund Heng and Antony Licata to use actions of fusion categories to generalise this to non-abelian groups.

Categories of relations including Weinstein's (linear) symplectic category are known to provide semantics convenient for quantum theory. We apply this philosophy to quantum \(L_\infty\) algebras, which give a homotopy algebraic framework for perturbative quantum field theories. Using homological perturbation theory to formalize a finite-dimensional incarnation of Batalin-Vilkovisky path integrals, we introduce a categorical perspective on quantum \(L_\infty\) algebras generalizing the minimal model theorem. No knowledge of QFT, symplectic geometry or homological perturbation theory will be assumed. This is a joint work with Ján Pulmann and Branislav Jurčo in progress.