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
******************************************************
M. Bunge: Locally Quasiconnected Toposes
******************************************************
G. Lukacs: Duality Theory of Locally Precompact Groups
***************************************************
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.
*************************************************************
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.
*************************************************************************
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.
*****************************************************************
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.
******************************************************************
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.
****************************************************************
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.
******************************************************************
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.
*************************************************************
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.
*************************************************************************