Tag - Category theory

Christopher Dean: Globular Multicategories with Homomorphism Types

28th May, 2020

We introduce structures called globular multicategories with homomorphism types. We discuss how various collections of "higher category-like" objects can be used to to construct these globular multicategories. We show how to obtain a number of higher categorical structures using this data. We will see that in this setting there is a precise sense in which:

  • types are higher categories,
  • dependent types are profunctors,
  • terms are higher functors,
  • terms in a dependent context are higher transformations,
  • there is a higher category of all types and terms.

James East: Presentations for tensor categories

15th May, 2020

Many well-known families of groups and semigroups have natural categorical analogues: e.g., full transformation categories, symmetric inverse categories, as well as categories of partitions, Brauer/Temperley-Lieb diagrams, braids and vines. This talk discusses presentations (by generators and relations) for such categories, utilising additional tensor/monoidal operations. The methods are quite general, and apply to a wide class of (strict) tensor categories with one-sided units.

Steven Sam: Noetherian properties in representation theory

18th August, 2016

I’ll explain some recent applications of 'categorical symmetries' in topology, algebraic geometry, and group theory. The general idea is to find an action of a category on the object of interest, prove some niceness property (like finite generation), and then deduce consequences from the general properties of the category.

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.