The ATCAT Seminar is a seminar on topics in category theory, running since 1973.
The current ATCAT organizers are Cameron Krulewski and Luuk Stehouwer.
Regular time: Tuesdays at 2:30 PM Atlantic time
Location: Chase Building, Room 319
This is a hybrid seminar. Email ckrulewski [at] dal [dot] ca to be added to the seminar mailing list and to get the Zoom link.
September 9: Luuk Stehouwer, Dalhousie University
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.
September 16: Robert Raphael, Concordia University
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
September 30: Roberto Hernández Palomares, University of Waterloo
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.
October 7: Cameron Krulewski, Dalhousie University
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.
October 14: Bryce Clarke, Tallinn University of Technology
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.
October 21: Daniel Teixeira, Dalhousie University
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.
October 28: Brett Hungar, The Ohio State University
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.
November 18: David Kern, KTH Royal Institute of Technology
Iterated spans in higher-categorical exactness
Exactness for 2-categories is known to concern the effectivity of internal categories whose underlying graphs—seen as spans—define discrete two-sided fibrations. I will explain how a study of structures in iterated spans leads naturally to a generalisation for (∞,ℓ)-categories: the key ingredients are an identification of monads in iterated spans as internal higher categories, and the expressivity of (ℓ-1)-Cat-weighted (co)limits to both define higher-categorical kernel/quotient systems and transport iterated discrete two-sided fibrations to codiscrete two-sided cofibrations while eschewing the use of lax structures.
November 25: Sarah Meng Li, University of Amsterdam
Fault-tolerant State Preparation Using Fault-Equivalent ZX Rewrites
Logical state preparation is a foundational task in fault-tolerant quantum computing, as it underpins not only algorithm execution but also magic-state distillation and syndrome extraction. Recently, Rodatz, Poór, and Kissinger (RPK) introduced a powerful framework for designing quantum circuits that are fault tolerant by construction, offering an alternative to the conventional brute-force approach. Using the ZX calculus as a diagrammatic language, they begin with idealised gadget specifications and apply a sequence of provably sound transformations to obtain fault-tolerant implementations.
In this talk, we explore two concrete applications of the RPK framework to logical state preparation. We start by introducing the key concepts and diagrammatic tools needed to reason about fault tolerance. Using fault-equivalent ZX rewrite rules, we demonstrate how to reproduce Goto’s construction for preparing logical stabiliser states in the Steane code. Finally, we propose resource-efficient fault-tolerant state-preparation circuits for the [[8, 3, 2]] colour code.
December 2nd: Hayato Nasu, Dalhousie University
On the decomposition of a strong epimorphism into regular epimorphisms
There are several different ways to generalize the notion of surjections to general categories. Surjections are the functions that do not factor through any proper subset of the codomain, which leads to the concept of strong epimorphisms. At the same time, surjections are the functions obtained by taking quotients of the domain, which are abstracted as regular epimorphisms. The two classes of morphisms coincide in regular categories, such as the categories of sets, groups, rings, or other algebras, but not in general. Still, it is known that a strong epimorphism in locally presentable categories can be expressed as a transfinite composite of regular epimorphisms.
I will discuss the problem of how close the two classes in a given category are, specifically, how many regular epimorphisms are required for this expression. Our approach is based on two syntactic presentations of locally finitely presentable categories, namely partial Horn theory and generalized algebraic theory, which I will explain in the talk.
This is joint work with Yuto Kawase.