Edinburgh category theory seminar

Under suitable smallness and completeness conditions, a functor \(F \colon \mathcal{C} \to \mathcal{D}\) induces a functor \(F_\# \colon \mathsf{Mnd}(\mathcal{C}) \to \mathsf{Mnd}(\mathcal{D})\) between the respective categories of monads. It sends a monad \(T\) to the right Kan extension of \(FT\) along \(F\); this is the pushforward of \(T\) along \(F\). In this talk I will introduce the construction and all the players involved, and give many of its interesting properties.

There will be examples throughout, especially of pushforwards along the inclusion \(\mathsf{FinSet} \to \mathsf{Set}\). In particular, I will give an easy answer to the following question: what endofunctors of the category of compact Hausdorff spaces fix underlying sets?

This talk is based on a recently updated preprint: https://arxiv.org/abs/2406.15256

Magnitude can be thought of as the canonical measure of the size of an enriched category, under suitable conditions. But in this talk, I will emphasize the magnitude not of categories but of set-valued *functors*, and introduce a new, dual concept: the "comagnitude" of a set-valued functor. Where magnitude is additive and closely related to colimits, comagnitude is multiplicative and closely related to limits. This is work in progress and there are many questions, but along the way, I will describe at least one definite result: a new way of deriving the concept of entropy from magnitude.

I will not assume any previous knowledge of magnitude. Details can be found in a pair of n-Category Café blog posts, beginning with https://golem.ph.utexas.edu/category/2025/01/comagnitude_1.html.

After briefly introducing (higher) topoi we'll have a look at how (co)monads can be used to study questions of descent and group actions in (higher) category theory. With these tools in hand we can define a very general notion of monodromy inside a topos and come to a classification theorem similar to that of monodromy for topological spaces. Though this monodromy theorem uses only tools from relatively basic category theory it is able to recover deep theorems in the world of chromatic homotopy theory, which I will explain very briefly at the end (no knowledge of chromatic homotopy theory will be required).

Infinity-operads are gadgets for encoding homotopy-coherent algebraic structures. By a variant of the envelope functor, one can identify infinity-operads with certain symmetric monoidal infinity-categories equipped with a monoidal functor to FinSet. This result is due to Haugseng and Kock, and in my talk I will describe the proof and discuss some implications.

The category of Hilbert spaces and contractions, the category of sets and partial injective functions, and various categories of probability spaces and stochastic maps, look like categories of relations, and yet do not fit into the established theory of relations in regular categories. To explain these anomalous examples, I will introduce the new notion of regular-ish independence category (building on Simpson’s notion of independent pullback), and develop the theory of relations in these categories. In particular, I will describe a bijective correspondence between regular-ish independence categories and dagger categories with dilators (a concept I introduced in my last talk at the seminar) that parallels the well-known correspondence between regular categories and tabular allegories. Regular-ish independence categories are sent to their associated dagger category of relations, while dagger categories with dilators are sent to their wide subcategory of coisometries.

In this talk I'll give an overview of an emerging picture in the study of directed graphs, and of various homology theories that have been attached to them. This will centre on the magnitude-path spectral sequence, due to Asao, which unites the two known theories of magnitude homology (which categorifies Leinster's notion of magnitude of a graph) and path homology (also known as GLMY homology, which has an important homotopy-invariance property). This development has sparked a flurry of new work trying to understand the spectral sequence and its properties. I'll try to highlight some of this recent work, including contributions of Emily Roff and myself, and I'll also discuss some interesting future challenges.

I will give an overview of how monads are used in programming to capture computations with side effects. Originally introduced by Eugenio Moggi, the Kleisli category of a (strong) monad specifies the semantics of an impure computation that may produce side effects, such as exceptions or outputs. Further, I will present the opposite version of this picture which studies notions of comonadic computations. Comonads can be used to specify programs that are context-dependent: their meaning depends on the environment they are placed in.

I will compare the two notions and demonstrate how they can interact. Throughout the talk, there will be many examples.

Many people working in graph theory today are interested in thinking of graphs as analogous to spaces, and developing homotopy theories for graphs. One particular categorical point of view on this idea can be found in the work of Lawvere, dating back to the 1980s. Lawvere observed that some elementary toposes seem to deserve to be thought of as 'categories of spaces', and he attempted to axiomatise toposes of this type in his work on cohesive toposes; illustrating the theory with the example of two distinct categories of graphs. In this talk we will explain Lawvere's ideas and discuss a particular homotopy theory for graphs that emerges from Lawvere's axioms.

Algebraic theories, such as the theories of groups, rings, and modules over a given ring, are often studied category theoretically as finitary monads on \(\mathbf{Set}\). Which such theories have the property that all their algebras are free? It turns out that we can write them all down relatively easily! Proving this result, however, is a different matter. One proof was given by Kearnes, Kiss, and Szendrei, in the language of classical universal algebra; Tom Leinster and I are working to build an understanding of this result in the language of monads. I will talk about what these special theories are, some of the properties they share, and some of the ways we're investigating the result in question.

A 2-group is a groupoid with a structure analogous to that of a group. By a representation of a 2-group \(G\) I mean a (weak) 2-functor from \(G[1]\), the one-object 2-groupoid with \(G\) as 2-group of self-equivalences of the unique object, to some target 2-category \(\mathcal{C}\). Taking the 2-category of Baez-Crans 2-vector spaces as \(\mathcal{C}\) leads to a bad representation theory because we lose almost all the information about the 2-group. In this talk I'm going to consider a larger 2-category of 2-vector spaces and show that the 2-category of Baez-Crans 2-vector spaces indeed embeds within it. The objects in this new 2-category should be classified by some cohomology of which I only know the cocycles and coboundaries at degree three. The talk will be a report of work in progress.

Partial Boolean algebras (pBAs) were introduced by Kochen and Specker in their seminal work on contextuality, a key signature of non-classicality in quantum mechanics which has more recently been linked to quantum computational advantage. They provide a natural (algebraic-)logical setting for contextual systems, an alternative to traditional quantum logic à la Birkhoff–von Neumann in which operations such as conjunction and disjunction are partial. In the key example of the projectors on a Hilbert space, the operations are only defined in the domain where they are physically meaningful, namely for commuting projectors, which correspond to properties of the quantum system that can be tested simultaneously. Contextuality manifests in that forcing totality leads to a contradiction, by collapsing to the trivial one-element Boolean algebra where \(0 = 1\).

In this talk, as a first step in the study of Stone-type dualities in this partial-algebraic setting, we extend the classical Tarski duality between sets and complete atomic Boolean algebras (CABAs). We establish a dual equivalence between the category of transitive partial CABAs and a category of exclusivity graphs, which are interpreted as spaces of possible worlds of maximal information with edges representing logical exclusivity. The classical case appears as the complete graphs, since all possible worlds are mutually exclusive. Correspondingly, morphisms are relaxed from functions (between sets) to certain kinds of relations (between exclusivity graphs). The result implies that any transitive partial CABA can be recovered from its graph of atoms as an algebra of cliques (sets of pairwise exclusive worlds) modulo the identification of cliques that jointly exclude the same set of worlds.

This ‘quantum’ extension of the simplest Stone-type duality reveals a connection between Kochen and Specker’s algebraic-logical setting of partial Boolean algebra and modern approaches to contextuality. Under the duality, a transitive partial CABA witnessing contextuality, in the Kochen–Specker sense of having no homomorphism to the two-element Boolean algebra, corresponds to an exclusivity graph with no ‘points’, i.e. with no maps from the singleton graph, equivalently described as stable, maximum-clique transversal sets.

(Joint work with Samson Abramsky)

This is a talk about something categorical done non-categorically, about a story more than half a century old that seems never to have been told. Sets as conceived by professional set theorists are quite unlike sets as used by other mathematicians. In traditional set theory, every element of a set is again a set, but no one else views the real number 6 as a set; and in that tradition, it always makes sense to ask whether one set is an element of another, whereas in practice, one would never ask whether the 2-sphere is an element of the trivial group. In the 1960s, Lawvere proposed axioms for set theory much better suited to mathematical practice. Although his axioms became well known in topos theory, it has been completely underappreciated that they can be stated, and set theory developed on top of them, without mentioning the word "category" (let alone "topos") even once. This talk is a warm-up for this semester's Axiomatic Set Theory course, where I will do exactly that.

The theory of contractions on a Hilbert space plays an important role in modern functional analysis. It is built upon Sz.-Nagy's unitary dilation theorem, which says that every contraction on a Hilbert space admits a minimal unitary dilation (a unitary dilation of a contraction \(T \colon X \to X\) is a unitary \(U \colon Y \to Y\) on a Hilbert space \(Y\) containing \(X\) via an isometry \(M \colon X \to Y\) such that \(T = M*UM\)). This talk is about an abstraction of the notion of contraction to suitably nice \(*\)-categories, and will build to a category-theoretic proof of a variant of Sz.-Nagy's theorem.

Delta lenses are functors equipped with a functorial choice of lifts, and specialise to split opfibrations when these lifts are opcartesian. In this talk, I will reflect on some of the many similarities between the rich theory of split opfibrations and the growing theory of delta lenses, and outline some yet unanswered questions. In particular, I will explore possible ways to prove that the category of split opfibrations forms a reflective subcategory of delta lenses.

We study the concept of the \(\ell_p\)-Vietoris–Rips simplicial set and the \(\ell_p\)-Vietoris–Rips simplicial complex of a metric space, where \(1\leq p \leq \infty\). For \(p=\infty\) we obtain the classical theory of Vietoris–Rips complexes, and for \(p=1\) we obtain the simplicial set whose homology is the blurred magnitude homology. We prove several results that were known for the Vietoris–Rips complex in the general case: (1) we prove a stability theorem for the interleaving distance between the corresponding persistent modules; (2) we show that for a compact Riemannian manifold and a small enough parameter the homotopy type of all the "\(\ell_p\)-Vietoris–Rips spaces" coincide with the homotopy type of the manifold; (3) we show that the \(\ell_p\)-Vietoris–Rips spaces are invariant under taking the metric completion. We also prove that the \(n\)-skeleton of all of these constructions defines the same pro-simplicial set. In particular, the limit of the homology groups, when the parameter tends to zero, does not depend on \(p\).

The notion of a 2-Segal set encodes an algebraic structure that is similar to that of a category, but for which composition need not exist or be unique, yet is still associative. The fact that such structures give rise to Hall algebras, generalizing constructions in representation theory and algebraic geometry, is one of the primary motivations for studying them. In this talk, we look at 2-Segal sets that arise from trees and graphs and their associated Hall algebras. These examples are discrete versions of the analogous 2-Segal spaces of trees and graphs developed by Gálvez-Carrillo, Kock, and Tonks, and provide a way to explore various general constructions quite explicitly. Much of this work is joint with Borghi, Dey, Gálvez-Carrillo, and Hoekstra Mendoza.

A concrete category is one with a faithful functor to the category of sets; this is an incredibly weak sense in which a category can be thought of as being encoded by sets with extra structure and functions between them. Freyd proved that the homotopy category of pointed topological spaces is not concrete, a fact with philosophical and somewhat practical implications for Homotopy Theory. We will give an outline of the proof of this result and if there's time discuss some more recent work on generalising this result in the context of model categories.

Complement-reducible graphs, or cographs, are the class of graphs generated from a single-vertex graph through operations of disjoint union and complementation. In fact, any cograph may be expressed canonically as a sequence of these operations, and as such cographs naturally lend themselves to efficient algorithms in computing with notable applications in bioinformatics. But why should we care about them in category theory?

This talk will introduce you to the fascinating connection between cographs and iterated monoidal category theory. Building on intuition from the familiar category \(\mathbb{F}\) of finite sets, as the free symmetric 1-fold monoidal category on a commutative monoid, we will use category of cographs, \(\mathbb{D}\), to introduce us to the world of 2-fold monoidal and duoidal categories.

The path signature is a characterisation of paths which preserves the underlying concatenation structure. It has been used in stochastic analysis to define rough paths, and in machine learning to provide features for time series data. This has recently been generalised to the surface signature: a representation of parametrised surfaces (on rectangular domains) which preserves the underlying double groupoid structure of horizontal/vertical concatenations. In this talk, we discuss the path and surface signatures from a category-theoretic perspective in the piecewise linear setting. Based on ongoing work with Camilo Arias Abad (Universidad Nacional de Colombia en Medellín) and Francis Bischoff (University of Regina).

An indexed category is a (pseudo-)functor \(S : I \to \mathbf{Cat}\). In the theory of open dynamical systems, we get indexed categories where the domain is a category whose objects represent 'interfaces' and whose morphisms are ways to connect interfaces. Then \(S(i)\) is the category of 'systems with interface shaped like \(i\)', for some flavour of system determined by \(S\). An animated category is a bicategory that results from taking a category enriched over interfaces \(I\) and changing the enrichment along \(S\).

In my talk I will illuminate this construction with some examples, following Toby St Clere Smithe's blog post https://tsmithe.net/p/animating-cats.html.

Guarded recursion is a framework allowing for a formalisation of streams in classical programming languages. The latter take their semantics in cartesian closed categories. However, some programming paradigms do not take their semantics in a cartesian setting; this is the case for concurrency, reversible and quantum programming for example. In this talk, we focus on reversible programming through the prism of dagger categories, which are categories that contain an involutive operator on morphisms. After presenting classical guarded recursion, which takes its model in the topos of trees, we show how to introduce this framework into dagger categories. Given a dagger category, we build categories shown to be suitable for guarded recursion in multiple ways, via enrichment and fixed point theorems. Finally, we show that our construction is suitable as a model of reversible programming languages, such as symmetric pattern-matching.

Abstract: jww Ivan Di Liberti, https://arxiv.org/pdf/2403.15338

To include a wider audience, here are two perspectives on the topic of my talk.

Grothendieck toposes are sometimes described as generalized spaces, owing to the fact that a motivating example is the category of sheaves on a topological space and many properties of the space are reflected in the categorical structure in a way that can be generalized to other Grothendieck toposes. For sufficiently separated spaces (the so-called sober spaces, notably including all Hausdorff spaces), the construction of the category of sheaves is even a fully faithful embedding of spaces and continuous maps into the (2-)category of toposes and geometric morphisms, and points become geometric morphisms \(\mathbf{Set} \to \mathbf{Sh}(X)\). One consequence of this point of view is that one might expect to be able to apply pointwise reasoning to toposes, to understand the objects and constructs within a topos by examining what they look like at points. This turns out not to be possible in general! A topos is said to have enough points if any pair of distinct morphisms is distinguished by a point.

Grothendieck toposes classify theories in geometric logic, a fragment of infinitary first-order logic in which only finite conjunctions, set-indexed disjunctions and existential quantification are allowed in the construction of formulae (and axioms are sequents asserting that one such formula entails another). That a topos \(E\) "classifies" such a theory \(T\) means that the models of \(T\) in any Grothendieck topos \(F\) correspond to geometric morphisms \(F \to E\). In particular, set-theoretic models correspond to morphisms \(\mathbf{Set} \to E\), which is to say points of \(E\). The classifying topos of \(T\) having enough points exactly says that \(T\) is complete with respect to its set-valued models.

Some sufficient conditions for a topos to have enough points emerged fairly early in the development of topos theory (Deligne, Makkai-Reyes), but the extent to which they are related to one another or could be extended was not apparent. In this talk I present the state of the art, alongside some examples indicating that improving our results will be tricky!

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 tannakian duality. This begins as a pleasant relationship between groups and their categories of representations, but it expands to become a fundamental general categorification/decategorification duality. We will pile on increasingly sophisticated layers of structure, arriving eventually at some as-yet-unproved forms of this duality.

In the presence of a monoidal right adjoint \(G \colon \mathcal{V} \to \mathcal{U}\) between locally finitely presentable symmetric monoidal categories, we examine the behavior of \(\mathcal{V}\)-Grothendieck topologies on a \(\mathcal{V}\)-category \(\mathcal{C}\), and that of their constituent covering sieves, under the change of enriching category induced by \(G\). We prove in particular that when \(G\) is faithful and conservative, any \(\mathcal{V}\)-Grothendieck topology on \(\mathcal{C}\) corresponds uniquely to a \(\mathcal{U}\)-Grothendieck topology on \(G_*\mathcal{C}\), and that when \(G\) is fully faithful, base change commutes with enriched sheafification in the sense of Borceux–Quinteiro.

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.