Short Courses in Category Theory

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Fiona Torzewska: Mapping class groupoids and motion groupoids

4th November, 2020

An LMS online course in mapping class groupoids.

A topological phase of matter is a physical system whose behaviour may be effectively described via a topological quantum field theory i.e. functor from cob to vect. The study of topological quantum field theories has applications in quantum computing but also involves a lot of beautiful mathematics which is interesting in its own right. 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 (spatial) dimensional phases of matter is then to look for generalisations of the braid group. Braid groups can be equivalently defined as the mapping class groups or as the motions groups of points in a disk, as well as in several other equivalent ways. In these lectures we will introduce generalisations of these two definitions. In each case we will show first that these give us groupoids and then that we can get back to the classical definitions by considering the endomorphisms of a single object. The mapping class groupoid is a simpler construction but is not in general the right notion to take when considering particles moving through space. We will construct a functor from the motion groupoid to the mapping class groupoid and hence see which cases we can study only the mapping class groupoid. We will use lots of examples to aid intuition and intend this talk to be accessible to those with minimal knowledge of topology.

Markus Land: L-theory of rings via higher categories

21st September, 2020

An online lecture course by the University of Münster in L-theory of rings.

We will introduce Witt groups and various flavours of L-groups and discuss some examples. We will then discuss a process called algebraic surgery. This process permits, under suitable assumptions, to simplify representatives in L-groups, and we will touch on two flavours (surgery from below and surgery from above). We will indicate how these can be used to show that various comparison maps between different L-theories are isomorphisms (in suitable ranges). Then we will go on and discuss three methods that allow for more calculations: Localisation sequences, a dévissage theorem, and an arithmetic fracture square. Using those, we will calculate the L-groups of Dedekind rings whose fraction field is a global field.

Yonatan Harpaz: New perspectives in hermitian K-theory

21st September, 2020

An online lecture course by the University of Münster in K-theory of forms.

In this lecture series we will describe an approach to hermitian K-theory which sheds some new light on classical Grothendieck-Witt groups of rings, especially in the domain where 2 is not assumed to be invertible. Our setup is higher categorical in nature, and is based on the concept of a Poincaré ∞-category, first suggested by Lurie. We will explain how classical examples of interest can be encoded in this setup, and how to define the principal invariants of interest, consisting of the Grothendieck-Witt spectrum and L-theory spectrum, within it. We will then describe our main abstract results, including additivity, localization and universality statements for these invariants and their relation to each other and to algebraic K-theory via the fundamental fibre sequence.

Ryo Takahashi: Thick tensor ideals of right bounded derived categories of commutative rings

10th August, 2016

Let R be a commutative Noetherian ring. Denote by D-(R) the derived category of cochain complexes X of finitely generated R-modules with Hi(X)=0 for i>>0. Then D-(R) has a structure of a tensor triangulated category with tensor product ⊗RL and unit R. In this series of lectures, we study thick tensor ideals of D-(R), i.e., thick subcategories closed under the tensor action by each object in D-(R), and investigate the Balmer spectrum Spc D-(R) of D-(R), i.e., the set of prime thick tensor ideals of D-(R). Here is a plan.
  •   We give a complete classification of the (co)compactly generated thick tensor ideals of D-(R), establishing a generalized version of the Hopkins--Neeman smash nilpotence theorem.
  •   We construct a pair of maps between the Balmer spectrum Spc D-(R) and the prime spectrum Spec R, and explore their topological properties.
  •   We compare several classes of thick tensor ideals of D-(R), relating them to specialization-closed subsets of Spec R and Thomason subsets of Spc D-(R).

If time permits, I would like to talk about the case where R is a discrete valuation ring. My lectures are based on joint work with Hiroki Matsui.