## LMS YaMCATS Category Theory and its Applications Lecture Series.

Welcome to the LMS YaMCATS Category Theory and its Applications Lecture Series webpage.

This lecture series ‘category theory and its applications’ is linked to the LMS funded Yorshire and Midlands Category Seminar (YaMCATS) network.

This lecture series consists of two topics which illustrate the power of categorical methods to other areas of pure mathematics, mathematical physics and computer science.

The first topic is on the theory of differential categories, to be presented by Jean-Simon Lemay (University of Oxford). The theory of differential categories uses category theory to study the foundations of differentiation. Differential categories have been able to formalize various aspects of differentiation, from the very basic foundational aspects of differentiation to the more complex notions of differential geometry.

The second topic is mapping class groupoids and motion groupoids, to be presented by Fiona Torzewska (University of Leeds).The lectures will explore the use of some higher dimensional categorical structures in the context of topological phase of matter, that is a physical system whose behaviour is effectively described by a topological quantum field theory.

**Starting on Wednesday 28 October 2020 11am (GMT) to 12pm (GMT) for 7 weeks**

Please register with the ICMS here, to receive future announcements and the zoom link.

Call details will be sent out **30mins before the start of the seminar**

This event is kindly funded by the London Mathematical Society.

### Programme of Lectures:

All lectures will be recorded and archived by the ICMS, links will be found on this page in due course.

**Jean-Simon Lemay **(University of Oxford): *The Theory of Differential Categories. *

Abstract: The theory of differential categories uses category theory to study the foundations of differentiation. Differential categories have been able to formalize various aspects of differentiation, from the very basic foundational aspects of differentiation to the more complex notions of differential geometry. In these lectures, we propose to introduce and summarize the theory of differential categories, as well as discuss interesting examples and applications. There will be five lectures, and a possible bonus one:

**Introduction (30 minutes)**: overview introduction to the theory of differential categories. We will discuss its history and motivations. In particular, we will take a look at the "map of differential categories" and discuss how each stage of the theory of differential categories are connected.

**Lecture 2) ** *Differential Categories* (45 minutes): Differential categories are axiomatized by the basic algebraic foundations of differentiation and provide the categorical semantics of differential linear logic.

**Lecture 3) ** *Cartesian Differential Categories *(45 minutes): Cartesian differential categories formalize the directional derivative and the theory of multivariable differential calculus over Euclidean spaces, and provide the categorical semantics of the differential lambdacalculus.

**Lecture 4) ***Restriction Differential Categories* (45 minutes): Restriction differential categories formalize differential calculus over open subsets. We will also introduce restriction categories.

**Lecture 5) ***Tangent Categories* (60 minutes): Tangent categories formalize tangent bundle structure and the theory of smooth manifolds.

**Bonus Lecture:** *Reverse Differential Categories* (45 minutes): Reverse differentiation is used in programming for efficient computations. We will discuss Cartesian reverse differential categories, a recent introduction to the theory of differential categories, which axiomatizes reverse differentiation, and their relationship to Cartesian differential categories, which axiomatize the forward derivative.

**Fiona Torzewska **(Univeristy of Leeds): *Mapping class groupoids and motion groupoids*

Abstract: I plan to do a series of three talks. By topological phase of matter we mean a physical system whose behaviour is effectively described by a topological quantum field theory. A central role in the description of topological phases of matter in 2 (spatial) dimensions is played by the representations of braid groups. A natural generalisation to study the statistics of higher dimensional phases of matter is then to look for higher dimensional generalisations of the braid group. Braid groups can be equivalently defined as the mapping class groups or as the groups of motions of points in a disk, as well as in several other equivalent ways. In these lectures we will introduce two generalisations. By fixing some ambient manifold $X$ and taking the homeomorphisms $f\colon X \to X$ which send a submanifold $A$ into a submanifold $B$, up to suitable isotopies, we can construct a mapping class groupoid. By instead taking paths in the space of selfhomeomorphisms of $X$ starting at the identity and ending on a homeomorphism which sends $A$ into $B$, up to a suitable notion of equivalence, we construct a motion groupoid. In each case we will show first that these give us categories and then that we can get back to the classical definitions by considering the endomorphisms of a single object. We will then look at a higher dimensional example for which the endomorphisms of a single object in the motion groupoid and the mapping class groupoid again coincide, the loop braid groups. Here the ambient space is the 3 ball and each group corresponds to an ncomponent trivial link. We will sketch the key ideas required to prove this equivalence in the case of the loop braid groups.