ATLANTIC CATEGORY THEORY SEMINAR

Details about the Atlantic Category Theory (ATCAT) Seminar for the 2019 winter term can be found below. Information about current seminars can be found here.

2019 WINTER TERM

• • January 8th, 2019.

  • Dominique Unruh (University of Tartu)
    Quantum Relational Hoare Logic

    Abstract

    We present a logic for reasoning about pairs of interactive quantum programs - quantum relational Hoare logic (qRHL). This logic follows the spirit of probabilistic relational Hoare logic (pRHL, Barthe et al. 2009) and allows us to formulate how the outputs of two quantum programs relate given the relationship of their inputs. pRHL was used extensively for computer-verified security proofs of classical cryptographic protocols. Since pRHL is not suitable for analyzing quantum cryptography, we present qRHL as a replacement, suitable for the security analysis of post-quantum cryptography and quantum protocols. The design of qRHL poses some challenges unique to the quantum setting, e.g., the definition of equality on quantum registers. We also implemented a tool for verifying proofs in qRHL and developed several example security proofs in it.

• • January 15th, 2019.

  • Robert Paré (Dalhousie University)
    The Double Category of Rings

    Abstract

    I will explain what double categories are and give some basic definitions. I will then explore the way these play out in the case of the double category of rings, i.e. the double category whose horizontal morphisms are homomorphisms and whose vertical ones are bimodules. ("Bimodules?", I hear you say! "Are they morphisms?" Well, that's one question I will address.)

• • January 22nd, 2019.

  • Toby Kenney (Dalhousie University)
    Abstract Convexity Spaces and Completely Distributive Lattices

    Abstract

    An abstract convexity space is a set X equipped with a chosen collection of convex sets that is closed under arbitrary intersections and directed unions. In this talk, I will discuss some background on abstract convexity spaces, provide some examples which spurred my interest in them, and show a connection with completely distributive lattices.

• • January 29th, 2019.

  • Geoffrey Cruttwell (Mount Allison University)
    A Linear/Non-Linear Approach to Tangent Categories (Part I)

    Abstract

    A tangent category is a structural setting for differential geometry. The central part of its structure is a functor T from a category C to itself, abstracting the tangent bundle endofunctor on the category of smooth manifolds.

    However, in differential geometry, the tangent bundle functor is often viewed as going between different types of objects: intuitively, it is seen as "providing a linear approximation to a non-linear object". Thus, one may wonder whether tangent categories should be axiomatized by starting with a functor between different categories (one of "non-linear" objects, one of "linear" objects), as opposed to starting with an endofunctor of a single category.

    In the first talk of this two-part series, I will review the definition of tangent categories and highlight how the existing theory defines linear objects. In the second talk, I will discuss work-in-progress on following the line of thought above to define a new "linear/non-linear" axiomatization of tangent categories, and describe some of the benefits and challenges this approach entails.

• • February 5th, 2019.

  • Geoffrey Cruttwell (Mount Allison University)
    A Linear/Non-Linear Approach to Tangent Categories (Part II)

    Abstract

    A tangent category is a structural setting for differential geometry. The central part of its structure is a functor T from a category C to itself, abstracting the tangent bundle endofunctor on the category of smooth manifolds.

    However, in differential geometry, the tangent bundle functor is often viewed as going between different types of objects: intuitively, it is seen as "providing a linear approximation to a non-linear object". Thus, one may wonder whether tangent categories should be axiomatized by starting with a functor between different categories (one of "non-linear" objects, one of "linear" objects), as opposed to starting with an endofunctor of a single category.

    In the first talk of this two-part series, I will review the definition of tangent categories and highlight how the existing theory defines linear objects. In the second talk, I will discuss work-in-progress on following the line of thought above to define a new "linear/non-linear" axiomatization of tangent categories, and describe some of the benefits and challenges this approach entails.

• • February 26th, 2019.

  • Jeff Egger
    Stratified Spaces and Tangent Categories: A Cautionary Tale

    Abstract

    We survey a number of notions of "stratified space", with the idea of obtaining a tangent category that has all pullbacks, but which is not so far removed from standard concepts of differential geometry as a model of synthetic differential geometry.

• • March 5th, 2019.

  • Michael Lambert (Dalhousie University)
    A Site for Continuous Group Actions

    Abstract

    In this talk, we will exhibit the well-known site for the category of continuous actions of a topological group. More precisely, we will discuss the sheaf category to which the category of topological group actions is equivalent. Time permitting, we will indicate how this might be generalized to a 2-site for continuous actions of 2-groups.

• • March 12th, 2019.

  • Robert Paré (Dalhousie University)
    Doubling Down on Morphisms of Rings

    Abstract

    This is continuation of the January 15th talk.

• • March 19th, 2019.

  • Marzieh Bayeh (Dalhousie University)
    Quantum Algorithms and Topological Data Analysis

    Abstract

    As a mathematician who is interested in different mathematical topics, from algebraic topology to quantum computation, it is my duty to find how the various branches of my mathematical knowledge can be linked to each other. In this talk I will give a summary of what I was reading on the relation between Quantum Computation and Topological Data Analysis (TDA). In particular, I will talk about a quantum algorithm that makes life easier for TDA.

• • March 26th, 2019.

  • Darien DeWolf (St. Francis Xavier University)
    A Comparison Lemma for Ehresmann Sites

    Abstract

    Given a functor u of essentially small sites, Kock and Meordijk's Comparison Lemma (1991) gives conditions on u which imply that the functor u* between their categories of sheaves is an equivalence. Using an equivalence between ordered groupoids and left cancellative categories, Lawson and Steinberg (2004) used the Comparison Lemma on the left cancellative categories side to present each étendue as the category of sheaves on some Ehresmann site (an ordered groupoid with some data topological in nature). As Pronk has recently reported, the equivalence between ordered groupoids and left cancellative categories can be extended to double categories. This talk will introduce a comparison lemma for Ehresmann sites which takes advantage of this extended equivalence, and allows us to avoid moving to left cancellative categories.

• • April 2nd, 2019.

  • Robert Raphael (Concordia University)
    It Follows by an Easy Induction (Part I)

    Abstract

    We give the first of two talks centered on the countable lifting property for partially ordered rings. The first talk will give three natural examples where the question makes good sense. We will mention the role of a proof by induction that made the topic of interest in the first place.

• • April 9th, 2019.

  • Robert Raphael (Concordia University)
    It Follows by an Easy Induction (Part II)

    Abstract

    We continue our discussion by studying rings which are rr-good. Algebraic examples are introduced followed by examples from rings of the form C(X). These "good" spaces include basically disconnected spaces and locally connected ones. A boundary condition is presented. We close with some open questions. Joint work with W.D. Burgess.

• • April 23rd, 2019.

  • Jonathan Gallagher (Dalhousie University)
    Learning is Smooth

    Abstract

    Differential programming languages give syntax for representing smooth functions on R^n. They facilitate learning algorithms: given inputs and expected outputs, the derivative can be used to figure out which component of a program needs to be tweaked.

    This talk will introduce differential programming and a semantics using tangent categories. Currently, an active area of research is how to extend differential programming beyond finite dimensions and how to allow learning with manifold domains. We will give a faithful semantics into the partial maps between micro linear spaces of a well adapted model of SDG. Such a model accommodates arbitrary dimensions, manifolds, and extra goodies.