ATLANTIC CATEGORY THEORY SEMINAR

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Details about the Atlantic Category Theory (ATCAT) Seminar for the 2018 fall term can be found below. Information about current seminars can be found here.

2018 FALL TERM

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• • September 11th, 2018.

  • Gabor Lukacs (Dalhousie University)
    Long Colimits of Topological Algebras

    Abstract

    Given a directed family (A_i) (i in I) of topological algebras (e.g., topological groups), their colimit can be taken in Top and in a category of topological algebras (e.g., Grp(Top)). When do these colimits coincide? It is known that if the indexing set I is the integers, then in most cases the colimit space topology fails to be an algebra topology (e.g., the binary operation is not continuous), and thus the two topologies are distinct. But what happens if the indexing set I is "long" in the sense that every countable subset has an upper bound? Do the two colimits always coincide? Joint work with Rafael Dahmen.

• • September 18th, 2018.

  • Dorette Pronk (Dalhousie University)
    A Double Category Approach to Ordered Groupoids and Left Cancellative Categories

    Abstract

    Ordered groupoids and left cancellative categories are two categorical concepts that have been used to describe partial symmetry. In a 2004-paper on the semi group forum Lawson constructed functors going back and forth between the category of ordered groupoids and the category of left cancellative categories. In my talk I will show that ordered groupoids can be viewed as a particular kind of double categories and how this can be used to show that Lawson's functors give rise to an equivalence of 2-categories. Joint work with Darien DeWolf.

• • September 25th, 2018.

  • Jeff Egger
    Generalised Hilbert Objects and Orthocomplemented Lattices

    Abstract

    Generalised Hilbert objects and orthocomplemented lattices Abstract: It is well-known that the closed subspaces of a Hilbert space form an orthocomplemented complete lattice. We investigate this fact from the point of view that there is a more general "closed subspaces" functor Ban --> Sup, and that both Hilbert spaces and complete orthocomplemented lattices can be regarded as objects in Ban and Sup (respectively) with extra structure. This leads us to a more general class of "closed subspaces" functors of the form V --> Sup, such that "generalised Hilbert objects" internal to V are always mapped to orthocomplemented lattices.

• • October 2nd, 2018.

  • Jeff Egger
    Positivity and Orthomodularity

    Abstract

    We present a definition of “generalised Hilbert object” (gHo), internal to a closed involutive (but not necessarily symmetric) monoidal category equipped with a compatible factorisation system. We show that complete orthomodular lattices can be construed as gHos in Sup, and that ordinary Hilbert spaces are gHos in Ban. We investigate which properties of the “closed subspaces” functor Q : Ban -> Sup entail that it preserves gHos.

• • October 16th, 2018.

  • Kohei Kishida (Dalhousie University)
    An Allegorical Semantics of Modal Logic (Part I)

    Abstract

    Modal logic has a very successful semantics called Kripke semantics, in terms of binary relations. This talk provides this semantics with a structural reformulation and extension, by taking advantage of the categorical approach to binary relations in terms of allegories. Our goal is two-fold: On the one hand, we will identify structural essences behind successful insights in the model theory of Kripke semantics, such as bisimulation invariance, correspondence theory, and duality theory. On the other, we will also show how Kripke semantics and these insights can be extended uniformly to relations that are based on not just sets and Boolean logic.

• • October 23rd, 2018.

  • Kohei Kishida (Dalhousie University)
    An Allegorical Semantics of Modal Logic (Part II)

    Abstract

    This is a continuation of the October 16th talk.

• • October 30th, 2018.

  • Marzieh Bayeh (Dalhousie University)
    The Svarc Genus of a Fibration

    Abstract

    The Svarc genus was defined by Svarc (Schwarz) in 1959. It is a number (integer) that is assigned to a fibration. This concept has been used to define new invariants, such as topological complexity and sectional category. In this talk, we will introduce the Svarc genus and discuss some of its properties.

• • November 6th, 2018.

  • Frank Fu (Dalhousie University)
    An Introduction to the Nominal Technique

    Abstract

    In pen and paper proofs, alpha-equivalence (e.g. forall x . P(x) is the same as forall y . P(y)) is not much of a concern, but it is one of the obstacles in formalizing meta-theoretical proofs in proof assistants (e.g. formalizing the meta-theory of programming languages). The so-called 'nominal technique' was first proposed by Gabbay and Pitts(1999). It is a mathematical theory (based on G-sets) intended to model and facilitate naming modulo alpha-equivalence. In this talk, I will introduce the category of nominal sets and identify some special objects and properties in this category.

• • November 20th, 2018.

  • Michael Lambert (Dalhousie University)
    Exactness of an Internal Colimit Functor for 2-Categories

    Abstract

    In this talk, an internal colimit functor for 2-categories will be defined. We shall see that the study of its exactness reduces to the exactness of the already well-understood internal colimit functor in ordinary topos theory.

• • November 27th, 2018.

  • Justin Makary (Dalhousie University)
    Completeness of the ZX-calculus for Stabilizer Quantum Mechanics

    Abstract

    The ZX-calculus is a graphical language for reasoning about quantum systems and processes which arose from categorical models of quantum mechanics. It is defined as a formal language equipped with rewrite rules which can be used to equate graphical representations. In this talk, I will show that the the rules of the ZX-calculus are complete for the so-called stabilizer fragment of quantum mechanics. This is done by showing that the graphical representations reduce to a normal form based on graph states and local Clifford operations. This talk is based on work by M. Backens.