Final Revised Schedule and Abstracts

All talks to be held in Fauteux Hall, Room 147A

Saturday, October 21st

8:45-9:15 Registration and Welcome

9:15-9:45 Gabor Lukacs (Manitoba): Duality theory of locally precompact groups

10:00-10:50 Break

10:50-11:20 Derek Wise (UC Riverside): Volumetric Field Theory

11:25-11:55 Eric Paquette (U. Montreal): The classical world from quantum theory

12:00-12:30 Benoit Valiron (Dalhousie): On a fully abstract model for a quantum linear lambda calculus

12:30-2:00 Lunch

2:00-3:00 Plenary Speaker: Ben Steinberg (Carleton): Ordered 2-complexes and inverse semigroups

3:00-3:30 Break

3:30-4:10 Jonathan Scott (Ottawa) : Operads and iterated loop spaces

4:15-4:55 Paul-Eugene Parent (Ottawa): Towards an adjoint to a Connes-Moscovici construction

5:00-5:40 Nick Gurski (Yale): Eckman-Hilton arguments in dimensions 1 and 2

Sunday, October 22nd

9:00-9:30 Bob Rosebrugh (Mount Allison): Implementing database design categorically (System demonstration)

9:35-10:05 Marta Bunge (McGill) : Locally quasiconnected toposes

10:10-10:40 Brian Redmond (Ottawa) : Soft linear logic

10:40-11:10 Break

11:10-11:40 Michael Winter (Brock): Cardinality in allegories

11:45-12:15 Robin Cockett (Calgary): Seely categories revisited

12:20-12:50 Jim Lambek (McGill) : Towards a Feynman category for the standard model.

Abstracts

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Nick Gurski

Title: Eckman-Hilton arguments in dimensions 1 and 2

Abstract: A special case of the standard Eckman-Hilton argument is that a monoid

(M, m,1) is commutative if and only if the multiplication m is a monoid

homomorphism when M x M is given the product monoid structure. A

similar phenomenon occurs in the context of monoidal categories, where

providing the tensor with the structure of a monoidal functor is the same as

giving a braiding for the monoidal category. I will discuss how this pattern

plays out in two contexts: moving from a braiding to a symmetry and monoidal

bicategories. This is joint work with Eugenia Cheng.

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J. Lambek

Title: In search of a Feynman category for the Standard Model

Abstract: An attempt is made to incorporate Feynman diagrams into a compact

*-autonomous category. The objects are finite multi-sets of pairs (x, a), x being a four

vector of real numbers representing a point in space-time, and a being a quadruple of

numbers -1, 0, +1, representing a fundamental particle. Its morphisms are generated by

motions (x,a)-->(y,a) , contractions (x,a)(x,b) --> (x,a+b) and expressions (x,a+b)-->(x,a)(x,b),

it being assumed that a_i = 0 or b_i = 0 or a_i + b_i = 0 for each i. It seems that the arrow

should denote a partial order, but then the compact Barr autonomous category degenerates into

a partially ordered group.

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Eric Paquette

Joint work with B. Coecke and D. Pavlovic

Title: The classical world from quantum theory

Abstract: We consider symmetric monoidal dagger-categories with

classical objects, that is, with dagger-Frobenius algebras. Such

categories have been identified to capture important fragments of

quantum behavior and, in particular, also quantum-classical

interaction including measurements. From any such category we recover

important subcategories namely: the (classical) probabilistic cone,

partial functions, relations, (doubly) stochastic maps, total

functions etc. We obtain an analysis of how two distinct instances of

classical probability live within the quantum structure:

superposition & mixture. We also recover important results from

quantum informatics, e.g. Nielsen's preorder on bipartite entangled

states, Naimarks theorem, dense coding etc.

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Bob Rosebrugh

Implementing database design categorically

(System demonstration)

Finite-limit, finite-sum (EA) sketches are the best syntactic structure

for modelling databases and their `views'. A series of articles with

Michael Johnson has explored and exploited this observation about EA

sketches and called it the Sketch Data Model (SkDM). This model extends

and enhances the standard ERA data model. The talk will begin with a brief

overview of these ideas.

Using Java to provide portability, we have written an application that

provides a user-friendly graphical design environment for EA sketches,

allows saving a design into an XML document and exporting that to a

database schema in SQL (the standard relational database language). The

application and some of its capabilities will be demonstrated.

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Ben Steinberg

Title: Ordered 2-complexes and inverse semigroups

Abstract:

Groups can be realized as fundamental groups of (combinatorial)

2-complexes; in fact a presentation of a group is really the same thing as

a 2-complex with a single vertex. A similar relationship exists between

groupoids and arbitrary 2-complexes. Many universal constructions, such

as amalgamated products and HNN-extensions, can be best understood in

terms of an analogous construction on 2-complexes.

Inverse semigroups can be viewed as a special kind of ordered groupoid.

One can define the notion of an ordered 2-complex and realize ordered

groupoids, in particular inverse semigroups, as the fundamental ordered

groupoid of an ordered 2-complex. Given an inverse semigroup

presentation, one can explicitly construct an ordered 2-complex that does

the job.

Applications include topological proofs of results concerning the finite

generation and presentability of maximal subgroups and the structure of

certain amalgamated products.

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Benoit Valiron

Title:

On a fully abstract model for a quantum linear lambda calculus

Abstract:

We study the linear fragment of the quantum lambda calculus, a

programming language for quantum computation with classical control

that was described in (Selinger, Valiron, 2006). We sketch the

language and discuss a categorical model. We also describe a fully

abstract denotational semantics in the category of completely

positive maps.

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Michael Winter:

Title: Cardinality in Allegories

Abstract: In this talk we want to introduce an internal notion of a

cardinality of a relation in the context of allegories. We

represent cardinal numbers by equivalence classes of objects and

introduce an operation mapping each relation to such an equivalence

class. The axiomatization of the cardinality function is based on a

stronger version of the Dedekind inequalities, introduced by

Y. Kawahara. We proof that for any tabular allegories there is a

unique cardinality function.

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Derek K. Wise

Title: Volumetric Field Theory

Abstract: In two dimensions, the quantum gauge theory pioneered by Yang and Mills is

known to be exactly solvable, and in fact "almost" a topological quantum

field theory - aside from the topology of a spacetime cobordism, it

requires only the total area as input. In this talk I show that in the

abelian case of Yang-Mills theory, also known as electromagnetism, this

fact remains true under categorification. More precisely, "n-form

electromagnetism" is a categorification of electromagnetism in which the

group U(1) is promoted to a strict abelian n-group. I describe n-form

electromagnetism in (n+1)-dimensional spacetime as a "volumetric field

theory" - a symmetric monoidal functor from the category of smooth

(n+1)-dimensional cobordisms with volume to the category of Hilbert

spaces.

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