This is the website for the ATCAT seminar fall 2025-winter 2026. The current site for Fall 2026 can be found here.
The ATCAT Seminar is a seminar on topics in category theory, running since 1973.
Regular time: Tuesdays at 2:30 PM Atlantic time
Location: Chase Building, Room 319 + Virtual
January 13: Robert Paré, Dalhousie University
Categories for the lazy mathematician
Strict double categories are category objects in Cat, and much of their basic properties follow from the general theory of category objects, where the results are proved using standard, albeit complicated, limit arguments. When proving such results, we secretly think that we are working with sets. Now, Cat has many set-like properties, but it is not Set, so we shouldn't be surprised when some things don't work. For example, profunctors don't compose properly.
Instead of pretending that categories are sets, we turn this on its head and pretend that sets are categories. Well, not sets, but the next best thing: the objects of a nice topos. Following Verity, we embed Cat in a topos that closely resembles it, whose objects we call lazy categories.
We investigate how much category theory carries over and what the advantages are of doing this.
January 20: Adrien DeLazzer Meunier, Dalhousie University
An introduction to reconstruction theory
Reconstruction theorems are a key tool in representation theory and theoretical physics. Details vary depending on the setting, but they all follow the same general idea; given some $\mathcal{V}$-enriched category $\mathcal{C}$, and a "fiber functor "$F\colon\mathcal{C}\to\mathcal{V}$ (these are like generalized forgetful functors), there is a canonical functor from $\mathcal{C}$ to the category of modules over the monoid $\text{End}(F)$. The reconstruction theorems give conditions under which this functor is an equivalence, and respects whatever structure $\mathcal{C}$ is assumed to have. Theorems of this sort are responsible for the zoo of definitions of "Hopf-like" algebras, as their categories of modules are used to characterize various flavours of category found throughout representation theory and physics.
In this lecture-style talk, I will give a sketch of reconstruction in the particularly nice context of fusion categories. The goal will be to understand the category-theoretic formalism—as this is consistent across many versions of reconstruction—and to see where and how the setting-specific details creep in. I will also gloss over a newer tool in the representation theory toolkit, known as de-equivariantization, and show how it can be understood through the lens of reconstruction theory.
January 27: Roman Geiko, UCLA
Invertible Stabilizer Codes
A stabilizer code is the common +1 eigenspace of a set of commuting unitaries drawn from a predetermined alphabet. While the most familiar choice of alphabet is the Pauli group, in this talk I will discuss its various generalizations. I will focus on one-dimensional codes defined as eigenspaces of a maximally commuting set of unitaries. From the many-body physics perspective, these codes are interesting because they can be realized as ground states of invertible Hamiltonians. Conjecturally, invertible many-body states (under suitable equivalence relations) are classified by invertible topological field theories. In this talk I will present a classification of invertible stabilizer codes up to finite-depth quantum circuits in terms of certain L-theory. Based on joint work with G. Shuklin.
February 3: Dorette Pronk, Dalhousie University
Orthogonal Factorization Systems for Double Categories
In this talk I will introduce a notion of orthogonal factorization system for double categories and give them the structure of a 2-category. We do this by showing that the operations of taking internal pseudo-categories in a 2-category and taking the pseudo-algebras for a 2-monad on a 2-category commute in the sense that they yield isomorphic 2-categories.
I will then give examples and showcase some of the properties of these factorization systems, such as their interaction with double fibrations.
February 10: Harshit Yadav, University of Alberta
Constructing Non-Semisimple and Non-Finite Modular Tensor Categories via Local Modules
Modular tensor categories (MTCs) that are finite and semisimple sit at a crossroads of several areas: they produce 3-manifold invariants via 3d topological field theory, they give representations of SL2(Z), and they arise from sources such as quantum groups and rational vertex operator algebras (VOAs). A broader goal, motivated for instance by non-rational (or logarithmic) conformal field theory, is to understand and construct MTCs without finiteness and semisimplicity assumptions, where questions like the existence of dual objects (rigidity), become more subtle. This talk describes a strategy to construct such non-semisimple and non-finite MTCs. Inspired by VOA extensions, it uses the category of local modules over a commutative ind-algebra object in a braided tensor category. Time permitting, applications to module categories of non-rational VOAs will also be discussed. This is based on joint work in preparation with Kenichi Shimizu.
February 24: Canceled due to inclement weather
March 3: Dan Berwick-Evans, UIUC
Topology and quantum fields
Structures in quantum field theory can have surprising mathematical depth, with a range of modern examples spanning multiple disciplines. This talk will give an introduction to a class of topological invariants arising from certain 2-dimensional quantum field theories. No background in quantum field theory will be assumed.
March 10: Martina Rovelli, UMass Amherst, University of Ottawa
From (∞,n)-categories to (∞,∞)-categories
There is an established notion of strict n-category and of (∞,n)-category for finite n. We will describe how there are two possibly competing ways of making sense of these for n=∞, depending on whether one takes the “inductive” or “coinductive” approach. The inductive and coinductive homotopy theories obtained from the strict homotopy theory of n-categories turn out to be equivalent, and are modeled by ω-categories. Instead, we will discuss how the inductive and coinductive homotopy theories obtained from the homotopy theory of (∞,n)-categories are not equivalent, as the latter is a proper localization of the former. This is joint work with Viktoriya Ozornova and Tashi Walde.
March 17: Scott Wesley, Dalhousie University
Towards Free Control Functors
Many models of computation, such as reversible circuits and quantum circuits, can be modeled formally using string diagrams in strict monoidal categories. The gates used to build these circuits are thought of as morphisms in a category with sequential composition corresponding to morphism composition and parallel composition corresponding to the monoidal product. Of course, one of the advantages of working with circuits is their combinatorial structure. To translate these combinatorial properties to the setting of monoidal categories, the primitive gates are thought of as generating morphisms in a monoidal signature (i.e., a one-object 2-polygraph). This freely defines a strict monoidal category of circuits, out of which it is easy to define syntactic transformations (endofunctors) and semantic interpretations (functors into typically large monoidal categories). The semantic soundness of a syntactic transformation can then be verified modulo a sound and complete equational theory.
One common syntactic transformation performed is the "control functor". This functor takes as input a circuit f, and produces a circuit C(f), which conditionally applies f. Unfortunately, this functor is not monoidal and does not lend itself well to the combinatorial methods afforded by viewing circuits as monoidal categories. On the other hand, these functors do respect monoidal products and braidings in a non-standard way. The goal of this talk is to determine whether control functors are determined by monoidal generators, and if so, what additional assumptions must be placed on the monoidal signatures so that they freely define control functors. This question will be answered from the point-of-view of generators and relations.
March 24: Natalia Pacheco-Tallaj, MIT
TQFTs and Anomalies
In this talk, I will describe some upcoming work with Basile, Krulewski, and Leone on anomaly cancellation problems arising in six-dimensional supergravity. I will begin by reviewing the notions of quantum field theory and quantum anomalies from a mathematical point of view, and explain how anomalies can be understood as invertible topological quantum field theories in one higher dimension. I will then give an overview of results relating the classification of invertible TQFTs to bordism groups of manifolds, and illustrate how tools from algebraic topology can be used to study anomaly TQFTs. The emphasis will be on a high-level and conceptual overview of the mathematical formalism, and no physics background is expected.
March 31: Aaron Fairbanks, Dalhousie University
Comonads as spaces
Comonads on Set generalize both categories and topological spaces. We characterize topological spaces as the density comonads of diagrams of subsets, i.e. topological subbases. Following Garner’s work on ionads, we develop aspects of theory of topological spaces for arbitrary comonads on arbitrary categories, including notions of basis, subbasis, continuous map, and specialization. Continuous maps and ordinary comonad morphisms form a double category, which in the case of comonads on Set corresponding to categories recovers the double category of functors and retrofunctors of Clarke and Di Meglio. Joint work with Kevin Carlson and David Spivak.
April 7: Naruki Masuda, Northwestern University
Categorical spectra and adjoints
Categorical spectra generalize spectra by allowing each level to be a pointed (∞,∞)-category rather than a space(=(∞,0)-category). After reviewing some basic facts, we will focus on what happens when we ask each level to have adjoints. We will in particular make some observations based on/related to the cobordism hypotheses.
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.