Atlantic Category Theory Seminar

Information about fall 2022 seminars can be found here.

Winter 2023 schedule

• January 24, 2023
  Christopher Dean, Dalhousie University
  
  Title: Strict Higher Modules
  
  Abstract:
  The monoids and modules construction on virtual double categories allows us to systematically construct collections of “category-like” objects, together with notions of functor, profunctor and natural transformation. I will discuss this construction and introduce an analogous “higher modules" construction for strict higher categories.
• January 31, 2023
  Geoff Vooys, Dalhousie University
  
  Title: Equivariant Categories, Functors, and Sheaves on Varieties
  
  Abstract:
  In algebraic geometry and in the local Langlands Programme (for p-adic groups, anyway), it is of fundamental importance to have a theory of equivariant sheaves, equivariant perverse sheaves, and an equivariant derived category in which both categories of sheaves exist as hearts of various t-structures. However, most formal properties of these categories and interactions have been studied in an ad hoc fashion and involve a grab bag of disparate and complicated tools. In this talk I will show that this need not be the case, and that there is a formalism that defines what it means to be an equivariant category in a very general sense that allows us to uniformly capture all three of the categories listed above as well as many, many more that arise both in practice and in abstract settings. We also, depending on time and interest, will be able to see how to define various flavours of equivariant functors, natural transformations, and adjoints, as well as what these allow us to do in practice.
• February 7, 2023
  Geoff Vooys, Dalhousie University
  
  Title: Equivariant Categories, Functors, and Sheaves on Varieties
  
  Abstract:
  In algebraic geometry and in the local Langlands Programme (for p-adic groups, anyway), it is of fundamental importance to have a theory of equivariant sheaves, equivariant perverse sheaves, and an equivariant derived category in which both categories of sheaves exist as hearts of various t-structures. However, most formal properties of these categories and interactions have been studied in an ad hoc fashion and involve a grab bag of disparate and complicated tools. In this talk I will show that this need not be the case, and that there is a formalism that defines what it means to be an equivariant category in a very general sense that allows us to uniformly capture all three of the categories listed above as well as many, many more that arise both in practice and in abstract settings. We also, depending on time and interest, will be able to see how to define various flavours of equivariant functors, natural transformations, and adjoints, as well as what these allow us to do in practice.
• February 14, 2023
  Matthew Yu, Perimeter Institute for Theoretical Physics
  
  Title: Anomalies for noninvertible surfaces and the 2-Deligne theorem [slides]
  
  Abstract:
  I will explain a generalization of the notion of an anomaly for a symmetry to a noninvertible symmetry. In particular I will focus on when the noninvertible symmetry is enacted by surface operators using the framework of condensation in 2-categories. I will introduce the concept of an anomaly for invertible symmetries. I will then give examples of when we have reasonable control of an anomaly for a noninvertible symmetry. This way of thinking allows us to prove theorems about the structure of the 2-category obtained by condensing a suitable algebra object. I will provide examples where the resulting category after condensation displays grouplike fusion rules and what the anomaly is in this case. I will then present the main result of this work, which categorifies a theorem of Deligne about the existence of fibre functors for 1-categories. This work was done in collaboration with Thibault Decoppet.
• March 7, 2023
  Frank Fu, Dalhousie University
  
  Title: Toward an induction principle for nested data types
  
  Abstract:
  A well-known problem in the theory of dependent types is how to handle so-called $\textit{nested data types}$. These data types are difficult to program and to reason about in total dependently typed languages such as Agda and Coq. In particular, it is not easy to derive a canonical induction principle for such types. Working toward a solution to this problem, we introduce $\textit{dependently typed folds}$ for nested data types. In this talk, I will be using the nested data type $\texttt{Bush}$ as a guiding example, I will show how to derive its dependently typed fold and induction principle in Agda.
  [Joint work with Peter Selinger]
• March 14, 2023
  Andre Kornell, Dalhousie University
  
  Title: The difference between bounded operators and binary relations
  
  Abstract:
  The category of Hilbert spaces and bounded operators and the category of sets and binary relations are two standard examples of dagger symmetric monoidal categories. They have many more properties in common. We will see two sets of axioms that characterize these categories up to equivalence. These axioms are category-theoretic, and these two sets have a large intersection.
• March 21, 2023
  Emily Cliff, Université de Sherbrooke
  
  Title: Smooth 2-groups and their principal bundles
  
  Abstract:
  A 2-group is a categorical generalization of a group: it's a category with a multiplication operation which satisfies the usual group axioms only up to coherent isomorphisms. In this talk I will introduce the category of Lie groupoids and bibundles between them, in order to provide the definition of a smooth 2-group. I will define principal bundles for such a smooth 2-group, and provide classification results that allow us to compare them to principal bundles for ordinary groups. As a consequence in specific settings, we obtain a line bundle over the moduli stack Bun_G(X) which is isomorphic to the Freed--Quinn line bundle from Dijkgraaf--Witten theory for a finite group G, This talk is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips. I will not assume any previous background on 2-groups, Lie groupoids, or Dijkgraaf--Witten theory.
• March 28, 2023
  Marcello Lanfranchi, Dalhousie University
  
  Title: Algebraic Deformation and tangent categories
  
  Abstract:
  

  The fundamental idea of deformation theory is to slightly modify an algebraic or a geometrical object so that the new one is still of the same kind of the original one, but its properties are different.

  For example, the deformation of an affine scheme corresponds to slightly changing the relations that define the corresponding coordinate ring.

  In this talk, we would like to present some interesting connections between algebraic deformation theory and tangent categories. We will show how to relate infinitesimal deformations of algebras to vector fields of a suitable tangent category. To explore this idea, we will make use of the theory of operads, by showing that the category (and also it's opposite one) of all algebraic operads over a fixed commutative and unital ring R, form a tangent category, naturally included in the tangent category of tangent monads over a tangent category.

  As part of my PhD research, this work is in collaboration with my supervisors Dorette Pronk and Geoffrey Cruttwell.

  I would like also to thank Geoff Vooys for the useful discussions I had with him on deformation theory.

• April 4, 2023
  Sarah Meng Li, Institute for Quantum Computing, University of Waterloo
  
  Title: Graphical CSS Code Transformation Using ZX Calculus [slides]
  Abstract:
  In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes (such as the Steane and quantum Reed-Muller codes) can be obtained from a common subsystem code. We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code transforming operations.