Meetings

Spring

Summer

Autumn

Winter

Travel expenses

To claim travel expenses:

Fifth meeting: 19 November 2024, University of Aberdeen

The meeting is open to everybody. Participants from the three nodes may be offered travel reimbursement. We hope to be able to reimburse all staff and PhD students from the nodes, but if (depending on participants numbers) any budget issues arise priority will be given to PhD students and early career researchers.

Location: The talks will be in Cruickshank G08. The coffee breaks and the lunch will be in the Mathematics common room in Fraser Noble (first floor). Here is a campus map.

Speakers:

Malin Altenmuller (University of Edinburgh)

Dorette Pronk (Dalhousie University)

Raffael Stenzel (University of Leeds)


Programme

Tuesday 19 November 2024

11:30-12:00 Welcome, tea and coffee
12:00-13:00 Dorette Pronk (Dalhousie University)
13:00-14:00 Lunch
14:00-15:00 Malin Altenmuller (University of Edinburgh)
15:00-15:30 EDI group discussion
15:30-16:00 Coffee break
16:00-17:00 Raffael Stenzel (University of Leeds)

18:00 -----  Dinner at Zizzi restaurant, Union Square, Aberdeen.

Title and Abstracts:

Dorette Pronk

Title: Covering Flatness for Sites with a Factorization System


Abstract: Grothendieck topoi are regular categories and hence have an epi-mono factorization system. In recent work, Darien DeWolf and Julia Ramos González and I have shown that this can be translated into the following property of sites: every Grothendieck topoi can be described as sheaves on a site with an orthogonal factorization system where all arrows in the left class are covering maps and all arrows in the right class are monics; we will call such sites CM-sites. Furthermore, there are enough CM-sites to describe the 2-category of topoi as the bicategory of left fractions to CM-sites with factorization preserving, covering preserving, covering-flat morphisms with respect to the Comparison Lemma maps.

 

Building on this result, we have extended the results from [1] and [2] and introduced double categorical sites for all topoi. We call these generalized Ehresmann sites and the idea is that the horizontal arrows in these sites are all coverings and the vertical part of the site is posetal and has assigned covering families of vertical arrows.

 

 When we take all functors that preserve the factorization systems between the CM-Sites and all double functors between the generalized Ehresmann sites, we obtain a 2-adjunction between the resulting 2-categories. This asjunction becomes a bi-equivalence if we restrict ourselves to the generalized Ehresmann sites where each vertical component has a maximal object. This provides the context to define maps of Ehresmann sites as those corresponding to the covering preserving and covering flat morphisms between CM-sites.

    The notion of being covering preserving is easy to translate, but the notion of covering flatness leads us to an interesting question: in the definition of covering flatness for Grothendieck sites with a factorization system, can we restrict our attention to diagrams indexed by a category that has a factorization system? 

    We will answer this question in the following generality: For any category C with either a strict or an orthogonal factorization system (E,M) and any diagram D: I --> C there is a factorization of D into a functor J: I --> \tilde{I} followed by a functor \tilde{D}: \tilde{I} --> C, where J is injective on objects and fully faithful, the category I has a strict factorization system and the functor\tilde{D} preserves the left and right classes. Furthermore, each cone over D has a unique extension to a cone over \tilde{D} (with the same vertex). The application to the notion of covering flatness for Ehresmann sites is quite interesting. There are  adjunctions between categories with factorization systems and double categories with certain properties presented in [3] that are different from the one we use to obtain generalized Ehresmann sites. However, in order to define covering flatness, we will use the functor from [3] to define an indexing double category induced by the category I, and we will see how both adjunctions work just in the right way to obtain a double categorical notion of covering flatness. 

    

I this talk I will present the construction for the factorization and its properties, and will discuss as much of its application to the morphisms between Ehresmann sites as time permits. 

 

References:

    

[1] D. DeWolf, and D. Pronk, A double categorical view on representations of \'etendues, Cah. Topol. G\'eom. Diff\'er. Cat\'eg. 61 (2020), no. 1, 3--56.

  

[2] M.V. Lawson, and B. Steinberg, Ordered Groupoids and \'Etendues, Cah. Topol. G\'eom. Diff\'er. Cat\'eg. 45 (2004), no. 2, 82--108.

    

[3] Miloslav \v{S}t\v{e}p\'an, Factorization systems and double categories, Theory and Applications of Categories 41 (2024), No. 18, 551--592.


Malin Altenmuller 

Title: A Category of Plane Graphs with Substitution and Pattern Matching


Abstract: String diagrammatic rewriting is an important technique to reason about morphisms in monoidal categories. It is typically realised by representing diagrams as open graphs together with a suitable notion of graph rewriting. For non-symmetric monoidal categories, these graphs have to be embeddable into the plane and hence must not contain any crossing edges.

I will present a category of plane graphs and their rewriting theory. The key feature of these graphs are distinguished boundary vertices which represent their (outer and inner) boundaries. Applying a graph rewrite rule is expressed by its action on these boundary vertices and we can prove that it preserves planarity.

I will show how graphs and substitutions yield an operad and introduce a notion of graph patterns as a co-operad. As both graphs and patterns live in the same underlying category we can study their interactions, which leads to a definition of graph pattern matching.


Raffael Stenzel

Title: “Higher covering diagrams”

 

Abstract: Descent properties are category-theoretic abstractions of the local-to-global principles considered in topology and geometry, as originally defined in Grothendieck’s work on the foundations of algebraic geometry. The talk will introduce a general descent property for functors between infinity categories, which will be illustrated with a review of standard descent properties and various examples from the literature. Using this descent property, I will then define the notion of a higher covering diagram, which is a generalisation of the notion of a universally effective-epimorphic cover. Finally, I present some results on higher covering diagrams and briefly outline a counterpart of this notion in Homotopy Type Theory.

 

1. Stenzel - Higher Geometric Sheaf Theories (https://arxiv.org/abs/2205.08646)

 Auxiliary references:

2. Anel, Joyal - Topo-Logie (https://doi.org/10.1017/9781108854429.007)

3. Janelidze, Tholen - Facets of Descent I (https://doi.org/10.1007/BF00878100)

 

Participants:


Fourth meeting: 1 October 2024, University of Edinburgh

The meeting is open to. Participants from the three nodes may be offered travel reimbursement. We hope to be able to reimburse all staff and PhD students from the nodes, but if (depending on participants numbers) any budget issues arise priority will be given to PhD students and early career researchers. 

Location: The meeting will take place in the Bayes Centre, in room G.03 on the ground floor.

Programme

11.00-11.30: Welcome with tea
11.30-12.30: Giacomo Tendas (University of Manchester)
12.30-13.30: Lunch
13.30-14.30: Carmen Constantin (University College London)
14.30-15.00: Lightning talks
15.00-15.30: Tea
15.30-16.30: Mark Lawson (Heriot-Watt University)
16.30-17.30: Social event

Further details will be announced later.

Titles and abstracts

Lightning talks


Giacomo Tendas: Regular theories from the (enriched) categorical point of view

In logic, regular theories are those whose axioms are built using only equations, relation symbols, conjunctions, and existential quantification. The categories of models of such theories have been widely studied and characterised in purely category theoretical terms through the notions of regular category and of injectivity class, that I will recall during the talk.

When moving to the context of enriched category theory, corresponding notions of "regular V-category" and "injectivity class" have been studied by several authors, but no enriched notion of regular logic was considered in the literature before. The aim of this talk, which is based on joint work with Rosicky, is to fill this gap by introducing a notion of "enriched regular logic" that interacts well with the category theoretical counterparts mentioned above. I will also explain how this is related to the internal logic of a topos, and that internal to Banach and metric spaces.
Background material:


Carmen Constantin: Commutation Groups and State-Independent Contextuality

I will start with an accessible and intuitive introduction to sheaf-theoretic contextuality, and talk about the three-level hierarchy introduced by Abramsky and Brandenburger. Then I will move on to discuss state-independent contextuality, as illustrated by the well-known Peres-Mermin magic square. This will give the motivation for introducing commutation groups in order to obtain a purely algebraic characterisation of state-independent contextuality. I will discuss how commutation groups can be presented by generators and relations and their connection to a directed version of the Heisenberg group. The main point of this construction is that it allows one to identify contextuality witnesses in the form of contextual words and I will give an overview of the settings in which such contextual words can arise. I will end by mentioning the unitary representations of commutation groups as subgroups of generalised Pauli n-groups.
Background material:


Mark Lawson: Non-commutative Stone duality
 I will show how classical Stone duality can be generalized to a duality between Boolean inverse monoids and a class of morphisms and a class of \'etale topological groupoids and a class of continuous functors.

I will discuss how this result might be generalized both on the arrow front and the object front which will lead me to state the result due to Garner linking  semigroup theory, category theory and theoretical computer science. Any undefined terms will be explained in the talk.
Background material:


Participants

Third Meeting: 3 May 2024, University of Aberdeen

The meeting is open to everybody. Participants from the three nodes may be offered travel reimbursement. We hope to be able to reimburse all staff and PhD students from the nodes, but if (depending on participants numbers) any budget issues arise priority will be given to PhD students and early career researchers. 

In view of the mini-excursion on Saturday morning, we will be able to offer a contribution towards overnight expenses to people from the nodes proportional to the number of attendees.

Location: The talks will be in Meston 1. The coffee breaks and the lunch will be in the Mathematics common room in Fraser Noble (first floor). Here is a campus map

Programme

Friday 3 May 2024

11:30-12:00 Welcome, tea and coffee
12:00-13:00 John Bourke (Masaryk University)
13:00-14:00 Lunch
14:00-15:00 Greta Coraglia (University of Milano)
15:00-16:00 Michael Johnson (UNSW Sydney)
16:00-16:30 Coffee break
16:30-17:30 Robert Booth (University of Edinburgh)

19:30 --- Dinner at Travancore

Saturday 4 May 2024 (morning)

Mini-excursion at Dunnottar Castle in Stonehaven. Stonehaven is a stop on the train line Aberdeen to Edinburgh, half an hour from Aberdeen. We will gather at the Stonehaven train station at 9:30 am, walk up to the castle and then come back to the village the same way in time for lunch (lunch will not be organized, but there are several places to eat in the village).

Local Information

Bars and Pubs:

CASC (especially good for whisky), The Wild Boar , The Prince of Wales

Ma Camerons , Tippling House (quite a fancy one).

Close to the restaurant: College Bar, McGynty's. 

Participants

Titles and abstracts

John Bourke: Analogies in the Bilax World
 This talk will be about the notions of skew monoidal category and algebraic weak factorisation system, which are two fairly recent categorical structures on which I have worked quite a bit yet still find somewhat mysterious.  I will remind the audience what they are, talk about why I find them interesting and explore some common features and curious analogies between them. 
The main prerequisites are a very basic knowledge of monoidal categories, monads and comonads.  Distributive laws between monads and comonads will play an important role in the talk.
Background material:


Greta Coraglia: A fibred perspective on the theory of (co)algebras

When you start looking for them, algebras and coalgebras appear everywhere: in modelling data types, in transition and dynamical systems, in all sorts of algebraic (!) structures such as monoids and modules, in convex and Banach spaces. In many of these cases, the interesting structure arises when one considers not only an endofunctor (or (co)monad), but a family of endofunctors that are compatible in a suitable way. We describe this behaviour and try to apply the theory of "families of things that are compatible in a suitable way", meaning the theory of Grothendieck fibrations, and see how far the switch in perspective can bring us.

Background material:

Michael Johnson: Higher Dimensional Associativity (with a little surprise)

(joint work with John Power)

Higher dimensional algebra (even for not very high dimensions like 2 and 3) comes with associativities in each dimension.  But a closer analysis of what associativity is really about, partly motivated by computer scientific considerations, leads to a deeper and more useful interpretation.  This deliberately accessible talk analyses that interpretation, shows that it has useful properties and that it delivers all that one could want at dimension 2, but that already by dimension 3, there is a little known extra delicacy that needs to be taken into account..

Robert Booth: A symplectic vision for the ZX-calculus
 For at least the past decade, there has been considerable development of
 sophisticated category theory using string diagrams to study dataflow in
 programs and quantum computations. This has led to the emergence of two
 successful research programs in diagrammatic reasoning: Graphical Linear
 Algebra and the ZX-calculus. Taking the parallel development of these
 two diagrammatic approaches seriously leads to a surprising connection
 via symplectic geometry. In this talk, I will introduce some elementary
 symplectic notions, describe a graphical language that unifies the two
 research programs and outline some current and future research directions.

Background material:

These build on the graphical languages described in these papers:

Second Meeting: 4 December 2023, University of Strathclyde

The meeting is open to everybody. Participants from the three nodes may be offered travel reimbursement. We hope to be able to reimburse all staff and PhD students from the nodes, but if (depending on participants numbers) any budget issues arise priority will be given to PhD students and early career researchers. Please register here.

Location: The talks will be in the McCance building in room MC301. 

Programme

10.30-11.00: Welcome with tea
11.00-12.00: Dan Marsden (University of Nottingham): "The Graphical Theory of Monads"
12.00-13.30: Lunch
13.30-14.30: Rita Fatimah Ahmadi (Imperial College London): "Bicategory of Topological Quantum Computation"
14.30-15.30: Neil Ghani (University of Strathclyde): "A Fibrational Approach to Differentiation"
15.30-16.00: Coffee Break
16.00-17.00: John Baez (University of California, Riverside): "Software for Compositional Modeling"

Participants:


Titles and Abstracts:

John Baez: "Software for Compositional Modeling"

Mathematical models of human interactions are important and widely used in epidemiology, but building and working with these models at scale is challenging.  I will explain two software tools for doing this, both based on category theory.   Modelers often regard diagrams as an informal step toward a mathematically rigorous formulation of a model.   Giving these diagrams a precise syntax using category theory has many advantages, but I will focus on those connected to "community based modeling": the process of working with diverse community members to build a model.   The next step is to tackle "agent-based models" and use them to help plan our response to climate change.


References:


Rita Fatimah Ahmadi: "Bicategory of Topological Quantum Computation"

Unitary Ribbon Fusion Categories (URFC) formalise anyonic theories. It has been widely assumed the same category formalises a model of topological quantum computation. However, we recently addressed and resolved this confusion and demonstrated while the former could be any fusion category, the latter is always a subcategory of \textbf{Hilb}. In this talk, I argue a categorical formalism which captures and unifies both an anyonic theory and corresponding models of topological quantum computation is a braided (fusion) bicategory.


References:



Neil Ghani: "A fibrational approach to differentiation"

With the growth of machine learning, there has been a renewed focus on differential geometry. In this talk I'll describe recent work on a fibrational unification of different differentiation doctrines.


References


Dan Marsden: "The Graphical Theory of Monads"

The formal theory of monads shows that much of the theory of monads can be developed in the abstract at the level of 2-categories. This means that results about monads can established once and for all, and simply instantiated in settings such as enriched category theory.

 

Unfortunately, these results can be hard to reason about as they involve more abstract machinery. In this talk, I shall present the formal theory of monads in terms of string diagrams --- a graphical language for categorical calculations. Using this perspective, I will show that many aspects of the theory of monads, such as the Eilenberg-Moore and Kleisli resolutions of monads, liftings, and distributive laws can be understood in terms of systematic graphical calculational reasoning.

 

The talk will serve as an introduction both to the formal theory of monads and to the use of string diagrams in non-trivial calculations, in particular, their application to calculations in monad theory. (Joint work with Ralf Hinze).


References


First meeting: 14 June 2023, University of Edinburgh 

The meeting is open to everybody. Participants from the three nodes may be offered travel reimbursement. We hope to be able to reimburse all staff and PhD students from the nodes, but if (depending on participants numbers) any budget issues arise priority will be given to PhD students and early career researchers. Please register here.


Programme

11.00-11.30: Welcome with tea
11.30-12.30: Vanessa Miemietz (University of East Anglia): "Categorification in Representation Theory"
12.30-13.30: Lunch
13.30-14.30: Elena di Lavore (Tallinn University of Technology): "Partial Markov Categories"
14.30-15.00: Panel discussion: "Encouraging women to apply for PhDs"
15.00-15.30: Tea
15.30-16.30: Nicola Gambino (University of Manchester): "Two-dimensional categorical logic"
16.30-17.30: Social drinks

The talks will be available over zoom at https://ed-ac-uk.zoom.us/j/83331873132 (Passcode: Noether99).

Vanessa Miemitz: "Categorification in Representation Theory"

Abstract: I will try to explain and motivate the ideas behind categorification in representation theory, before explaining in more detail some results on the specific example of Soergel bimodules, which categorify Hecke algebras.

Background material:

Elena di Lavore: "Partial Markov Categories"

Abstract: I will present partial Markov categories, giving an introduction to Markov categories and cartesian restriction categories. Markov categories encode stochastic processes and cartesian restriction categories encode deterministic processes with constraints. In the same way, partial Markov categories encode stochastic processes with constraints, observations and updates. In particular, we prove a synthetic Bayes theorem. This is recent joint work with Mario Román.

Background material: 

Nicola Gambino: "Two-dimensional categorical logic"

Abstract: Categorical logic, founded by Lawvere in the '60s, is generally concerned with the interplay between logic and  category theory, with applications in both directions. In recent years, motivation from various angles, including theoretical computer science, has led to first steps in the creation of two-dimensional categorical logic, in which ordinary set-based structures are replaced by category-based ones (e.g equivalence relations are replaced by groupoids), very much in analogy with the research program of categorification in algebra.

After reviewing the basics of categorical logic and outlining the key aspects of two-dimensional categorical logic, I will focus on an illustrative example, namely the 2-category of analytic functors, introduced in [FGHW] and studied further in [GJ]. This 2-category possesses a wealth of structure, thus giving a good indication of the potential and complexity of two-dimensional categorical logic. In particular, it provides a model of the differential lambda-calculus [ER], an extension of the lambda-calculus with a differential operator, in which it is possible to approximate lambda-terms by a form of the Taylor series expansion [ER2].

Background material:

Participants

Location

The meeting will take place in the Bayes Centre, in room G.03 on the ground floor.