# Meeting 28

## Thursday September 22, 2022

Hosted by the University of Cambridge

Supported by the London Mathematical Society

**Schedule: **

- 13:00-13:45: Lukas Heidemann (University of Cambridge)
- 13:45-14:30: Fiona Torzewska (University of Leeds)
- 14:30-15:00: Coffee break
- 15:00-15:45: Ming Ng (University of Birmingham)
- 15:45-16:30: Federico Olimpieri (University of Leeds)

**Location:**

Talks will be in the Computer Science department in room **SS03**. This is on the second floor, on the south side of the building. This part of the building requires swipe card access, please at reception and say you are here for the YAMCATS seminar they will let you through. The coffee break will be in the first floor common room.

The department is a half hour walk from the centre of Cambridge, or a one hour walk from the train station. There are e-scooters available to hire, and Uber operates in the city. There is a bus from the station but it is not very frequent. If you are driving you can park in the free Park & Ride car park, which is a 5-minute walk from the department.

**Titles and Abstracts:**

**Speaker: Lukas Heidemann **

**Title: Manifold Diagrams and the Geometry of Higher-Dimensional Categories**

**Abstract:** String diagrams are a graphical presentation of monoidal categories and 2-categories that capture algebraic coherence conditions as intuitive geometrical manipulations. While algebraic axiomatisations of 2-categories are merely inconvenient to work with, the coherence laws for weak n-categories become almost entirely impractical to deal with when n > 2. A geometrical approach to n-categories generalising string diagrams could make their usage practical but has so far only been achieved for n = 3. We present a geometric theory of manifold diagrams in arbitrary dimensions which correspond to terms in associative n-categories, a theory that we conjecture to be equivalent to that of weak higher-dimensional categories.

**Speaker: Fiona Torzewska**

**Title: Topological quantum field theories and homotopy cobordisms**

**Abstract: **In this talk I will discuss the construction of a category CofCos whose objects are topological spaces and whose morphisms are cofibrant cospans. Here the identity cospan is chosen to be of the form $X\to X\times [0,1] \xrightarrow X$, motivated by considerations regarding topological phases of matter. The category CofCos has a subcategory HomCob in which all spaces are homotopically 1-finitely generated. This category is potentially useful for studying representations of particle motion in topological phases of matter. I will discuss how representations of HomCob lead to representations of motion groupoids, and mapping class groupoids. Time permitting I will also give an explicit construction showing that a large class of topological quantum field theories extend to representations of HomCob.

**Slides**: here

**Speaker: Ming Ng**

**Title: Adelic Geometry via Topos Theory**

**Abstract:** In this talk, I will give a leisurely introduction to the theory of classifying toposes, before introducing a new research programme (joint with Steven Vickers) of developing a version of adelic geometry via topos theory.

To elaborate, let us highlight two important aspects of the story.

First, much of the theory-building in number theory has been guided by the following tension: completions of a number field ought to be treated in a symmetric way (cf. Hasse Local-Global Principle, product formula etc.) yet there also exists important differences between the Archimedean vs. non-Archimedean completions. This raises an important question: what is the right framework for us to understand this tension? In topos theory, our main point of leverage is that every topos classifies some (logical) theory T, and contains a “generic model” of T — which is “generic” in the sense that it freely generates all other models of T. In our programme, we ask: is there a topos of completions of the rationals Q? How might we go about constructing this topos? What can the generic completion tell us about the relationship between Archimedean vs. non-Archimedean completions?

Second, in order to work with classifying toposes we shall need to work “geometrically” — which effectively means pulling our mathematics away the set theory. This choice of methodology ends up revealing a deep nerve connecting topology and algebra, invisible from the perspective of classical mathematics. For instance, one important step of our project involves constructing the topos of places of Q, which incidentally provides a topos-theoretic account of the Arakelov compactification of Spec(Z). However, whereas the classical picture views the Archimedean place as a single point “at infinity”, our picture reveals that the Archimedean place resembles a blurred interval living below Spec(Z), raising challenging questions to our current understanding of the number theory.

This talk will discuss both aspects, along with some of their interesting implications.

**Slides**: here

**Speaker: Federico Olimpieri**

**Title: Categori****fying Graph Models of λ-calculus**

**Abstract: **Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalising the relational one, we identify the class of 'categorified' graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterisation of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have an isomorphic interpretation exactly when their Böhm trees coincide.