Atlantic Category Theory Seminar

Details about the winter 2021 Atlantic Category Theory (ATCAT) Seminar can be found below. Information about previous seminars can be found here.

Winter 2021 schedule

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• January 19th, 2021 [slides]
  Toby Kenney, Dalhousie University
  Stone Duality for Topological Convexity spaces
  Abstract

  A convexity space is a set X with a chosen family of subsets (called convex subsets) that is closed under arbitrary intersections and directed unions. There is a lot of interest in spaces that have both a convexity space and a topological space structure. In this talk, we will study the category of topological convexity spaces and extend the Stone duality between coframes and topological spaces to a duality between topological convexity spaces and sup-lattices. We will also identify some of the important classes of morphisms in the category of topological convexity spaces, and some of their properties, in preparation for identifying Euclidean spaces as object within the category of Topological Convexity Spaces.

• February 9th, 2021 [slides]
  Toby Kenney, Dalhousie University
  Euclidean Objects in the Category of Topological Convexity Spaces
  Abstract

  Last time, we studied the category of Topological Convexity spaces, and its relation to the category of sup-lattices. In this talk, we will look at how the classical Euclidean spaces, i.e. vector spaces over the real numbers, can be axiomatised in this Category.

• February 23rd, 2021 [slides]
  Dorette Pronk, Dalhousie University
  Orbispace Mapping Objects: Three Approaches, Two Results
  Abstract

  Orbispaces are defined like manifolds, by local charts. Where manifold charts are open subsets of Euclidean space, orbifold charts consist of an open subset of Euclidean space with an action by a finite group (thus allowing for local singularities). This affects the way that transition between charts need to be described, and it is generally rather cumbersome to work with atlases. It has been shown in [Moerdijk-P] that one can represent orbifolds by groupoids internal to the category of manifolds, with etale structure maps and a proper diagonal. We have since generalized this notion further and we now consider orbispaces as represented by proper etale groupoids in the category of locally compact, paracompact topological spaces (they will also be called orbigroupoids). Two of these groupoids represent the same orbispace if they are Morita equivalent.
  
  So we consider the bicategory of fractions with respect to Morita equivalences. For orbigroupoids G and H we can then consider the mapping groupoid [G, H] of maps and 2-cells in the bicategory of fractions. The question I want to address is how to define a topology on these mapping groupoids to obtain mapping objects for this bicategory. This question was addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers.
  
  I will approach this question from three different directions:
  
  1. When the orbifold G is compact, we can define a topology on [G,H] to obtain a topological groupoid OMap(G, H) so that Orbispaces(K × G, H) is equivalent to Orbispaces(K, OMap(G, H)). We will also show that OMap(G,H) represents an orbispace.
  
  2. For any pair of orbigroupoids G, H we can define a topology on [G,H] to obtain EMap(G,H) so that Orbispaces has the structure of an enriched bicategory: composition induces a continuous functor EMap(G,H) x EMap(H,K) --> EMap(G,K).
  
  3. There is a fibration structure on the category of orbigroupoids with groupoid homomorphisms as defined in [P-Warren]. (This can be derived from unpublished work by Colman and Costoya.) This implies that when G and H are stack groupoids, we may restrict ourselves to ordinary groupoid homomorphisms and their usual 2-cells.
  
  In this talk I will discuss the relationships between the topologies obtained in these ways, as well as the relationship with Chen's work. This is joint work with Laura Scull.
  
  [Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.
  [Moerdijk-P] I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), pp. 3-21.
  [P-Warren] Dorette A. Pronk, Michael A. Warren, Bicategorical fibration structures and stacks, Theory and Applications of Categories, Vol. 29, 2014, No. 29, pp 836-873.
  

• March 2nd, 2021 [slides]
  Nils Carqueville, University of Vienna
  Orbifolds of topological quantum field theories
  Abstract

  I will review the functorial approach to topological quantum field theories with defects, a generalised orbifold construction, and discuss several applications.

• March 9th, 2021 [slides]
  Geoffrey Cruttwell, Mount Allison University
  A categorical framework for gradient-based learning
  Abstract

  Many artificial intelligence systems use variants of the gradient descent algorithm to help them "learn". (For examples of such variants, see https://arxiv.org/abs/1609.04747). In this series of two talks, we'll see how many of these variants can be unified in a single categorical framework. The categorical tools we will use to build this framework include categories of parameterized maps, categories of lenses, and reverse derivative categories. The first talk will focus on introducing these three categorical structures, while the second talk will put the structures together and show how many of the gradient-based algorithms which are used in practice fit into the resulting framework.
  
  This is joint work with Bruno Gavranovic, Neil Ghani, Paul Wilson, and Fabio Zanasi.

• March 16th, 2021 [slides]
  Geoffrey Cruttwell, Mount Allison University
  A categorical framework for gradient-based learning
  Abstract

  Continuation of the previous seminar.

• March 23rd, 2021
  Martin Szyld, Dalhousie University
  The three F's in bicategory theory
  Joint work with P. Bustillo and D. Pronk
  
  Abstract

  We consider the notions of Fibration of categories, (pseudo)Filtered category, and the axioms for a category of Fractions. A basic fact involving them is: given a Fibration, if the arrows of the base category are (pseudo)coFiltered, then the cartesian arrows satisfy Fractions. This is a Proposition in SGA 4 (Exp. VI, Prop. 6.4) whose proof is left to the reader as an exercise, and I want to start this talk by solving this exercise. Let me tell you why.
  
  Each of the three "F" notions above has been considered for bicategories, or at least for 2-categories. I will start with what may be the easiest one to understand, that of Filtered: in a Filtered bicategory, in addition to asking for cones for two objects and for two parallel arrows, we add a third axiom asking for cones for parallel 2-cells. I will present the definitions of Filtered and pseudoFiltered bicategory, a set of axioms for a bicategory of Fractions, and some properties of Fibrations of bicategories that all fit this same pattern. We arrived at these notions when proving a "bicategory version" of the Proposition in SGA 4, in fact a small generalization.
  
  This result is part of an ongoing collaboration with P. Bustillo and D. Pronk, we're working on showing some basic properties of the bicategorical localization by fractions which are known in dimension 1. If time permits, I hope to mention how we ended up here within our current work and how this result can be applied here.

• March 30th, 2021 [slides]
  Abraham Westerbaan, Dalhousie University
  The structure of ω-complete effect monoids
  
  Abstract

  Effect monoids are a generalisation of {0,1} and [0,1], of Boolean algebras and the unit intervals [0,1] of commutative unital C*-algebras. Effect monoids appeared naturally in the study of effectuses: a type of category with finite coproducts 0, + and a final object 1 designed to reason about states s: 1⟶X and predicates p: X⟶1+1. When composing such a state and predicate, one gets a morphisms 1⟶1+1 that should be thought of as the probability that the predicate p holds in state s. It’s these morphisms 1⟶1+1 called scalars that form an effect monoid.
  
  Vanilla effect monoids are lousy structures: not much can be defined with(in) them, or be proven about them, while counter examples hard to find. This changes dramatically when the axiom of ω-completeness is added (that every ascending sequence in the effect monoid has a supremum.) Suddenly a rich and well-behaved structure emerges including division, lattice operations, and an abundance of idempotents. So well-behaved, in fact, that every ω-complete effect monoid can be represented as subspace of the continuous functions C(X,[0,1]) on a basically disconnected compact Hausdorff space X. For directed complete effect monoids we even get a proper categorical duality.
  
  This is based on joint work with Bas Westerbaan and John van de Wetering: https://arxiv.org/abs/1912.10040

• April 6th, 2021 [slides]
  Sarah Meng Li, Dalhousie University
  Reducing CNOT count in Clifford+T circuits on Connectivity Constrained Architectures
  Joint work with Vlad Gheorghiu, Michele Mosca, and Priyanka Mukhopadhyay
  
  Abstract

  While mapping a quantum circuit to the physical layer, one has to consider the numerous constraints imposed by the underlying hardware architecture. Connectivity of the physical qubits is one such constraint that restricts two-qubit operations like CNOT to “connected” qubits. SWAP gates can be used to place the logical qubits on admissible physical qubits, but they entail a significant increase in CNOT-count.
  
  In this talk we consider the problem of reducing the CNOT-count in Clifford+T circuits on connectivity constrained architectures. We “slice” the circuit at the position of Hadamard gates and “build” the intermediate portions. We investigate two kinds of partitioning – (i) partitioning the gates of the input circuit based on the locality of H gates and (ii) partitioning the phase polynomial of the input circuit. The intermediate {CNOT,T} sub-circuits are synthesized using Steiner trees, similar to the work of Nash, Gheorghiu, Mosca in 2020 and Kissinger, de Griend in 2019. Our algorithms have certain procedural differences that also help to further reduce the CNOT-count. In our experiment, we compared the performances of our algorithms while mapping different benchmark circuits as well as random circuits to some popular architectures like 9-qubit square grid, 16-qubit square grid, Rigetti 16-qubit Aspen, 16-qubit IBM QX5, 20-qubit IBM Tokyo. We found that for both the benchmark and random circuits our first algorithm using the simple slicing technique performs much better i.e. gives much less CNOT-count than the count obtained by using SWAP gates. Our second slice-and-build algorithm performs reasonably well for benchmark circuits.