We usually meet on Tuesdays at 2:30pm in room 319 of the Chase Building. The current organizers are Christopher Dean and Luuk Stehouwer. Information about summer/winter 2025 seminars can be found here.
On a generalization of dagger compact categories
The notion of a dagger compact category combines duals with an involutive dagger functor and provides a categorical setting for operator algebra and quantum theory. The key example of a dagger compact category is that of finite-dimensional Hilbert spaces. On the other hand, the symmetric monoidal category of finite-dimensional super Hilbert spaces is not dagger compact, yet naturally arises in quantum physics when fermions are present. In this talk I will provide a natural generalization of dagger compact categories to arbitrary rigid symmetric monoidal $n$-categories. The goal of this talk is not to overwhelm you with higher categories, but to spell out the definition in the case $n=1$ in full detail. The result will be a mild generalization of a dagger compact category that covers super Hilbert spaces as a special case.DL-closures and 2-3 closures applied to the ring $C^1(R)$.
In joint work with Barr and Kennison it was shown that commutative semiprime rings have a DL-closure and a 2-3 closure and that the ring of continuously differentiable real-valued functons is not closed in either sense. Our work is devoted to trying to describe the two closures of this ring. The methods are analytic often using basic ideas from calculus. A useful example sent by Alan Dow is presented.
Joint work with W. D. Burgess
Quantum graphs and spin models
Spin models for singly-generated Yang-Baxter planar algebras are known to be determined by certain highly-regular classical graphs such as the pentagon or the Higman-Sims graph. Examples of spin models include the Jones and Kauffman polynomials, as well as certain fiber functors. We will explore the notion of higher-regularity for quantum graphs as well as their potential to encode spin models. Time allowing, we will give examples of non-classical graphs with these properties.
Dagger Categories and Higher Spin Statistics
A functorial field theory is a symmetric monoidal functor from a bordism category to a target category. Using dagger categories, one can define a unitary functorial field theory to be a functor of dagger categories. In the invertible, fully-extended case, Lukas Müller, Luuk Stehouwer, and I used these ideas to prove a higher version of the spin-statistics theorem, which says that unitarity constrains the behavior of particles.
A second look at limits in double categories
Marco Grandis and Robert Paré introduced the study of limits in double categories, generalising weighted limits in 2-categories. They showed that a double category admits all limits indexed by double categories if and only if it admits products, equalisers, and tabulators. Unfortunately, there are many interesting limit-like constructions in double categories, such as restrictions, companions, conjoints, parallel limits, and local limits which do not arise as actual limits under this definition. In this talk, I will introduce the notion of limit indexed by a loose distributor, which captures all of these concepts as examples. The main theorem will be to show that a double category admits all limits indexed by loose distributors if and only if it admits parallel limits (= parallel products, equalisers, and tabulators) and restrictions. The talk will focus on exhibiting many examples of limits in the double category Span of sets, functions, and spans, and the double category Dist of categories, functors, and distributors. This talk is based on joint work with Nathanael Arkor.
Categories as directed spaces
The hypothesis that spaces are the same as ∞-groupoids has guided higher category theory for the past 20 years. Reversing this analogy, recent years have seen that constructions in homotopy theory are undirected shadows of their directed analogues in (∞,∞)-categories - for instance disks vs globes, Ω- vs categorical spectra, lax vs homotopy limits, Cartesian vs Gray products.
We will cover this philosophy and specialize in: "(∞,n)-categories are to (∞,∞)-categories as n-connected spaces are to spaces." We will approach this analogy via factorization systems.
Strings on tubes in 3-categories
Anchored planar algebras, introduced by Henriques, Penneys, and Tener, are an extension of Vaughan Jones' notion of planar algebras to arbitrary braided categories. The operadic composition of these algebras can be interpreted as a 3-dimensional graphical calculus of strings on tubes. We show that this graphical calculus can be recovered from the 3-dimensional graphical calculus of 3-categories. This allows for more general techniques to be applied to anchored planar algebras and expands the types of diagrams that anchored planar algebras can interpret.