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.

### 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 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 Set, whose algebras are the models for the theory in 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.