Category Theory Octoberfest 2006
                                        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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M. Bunge: Locally Quasiconnected Toposes

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G. Lukacs: Duality Theory of Locally Precompact Groups

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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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