Abstracts of talks received so far (revisions and additions
will probably be made)
=====================================
G Beaulieu
Title: Factorizing the Monad for Mixed Choice.
Abstract:
There has been growing interest in developing models for process calculi
which admit both a discrete probabilistic and a nondeterministic choice
operator. We will see that the axioms for mixed choice are a combination
of the axioms for discrete probabilistic choice and the axioms for
nondeterministic choice. Interestingly, the monad which captures the
correct behaviour for mixed choice is not the composition of the monads
for the separate choices. I will present a concrete definition for the
mixed choice monad and its factorization through the probabilistic and
the nondeterministic monads. Furthermore, I will point out interesting
categories arising from our monads which may help us lift to a
continuous probabilistic choice environment.
=====================================
Jeff Egger
Title: Linear distributivity and the Frobenius condition
Abstract: Recent work of Blute, Cockett and Seely suggests that the
Frobenius condition might be best understood in the context of
linear-distributive categories. But, I have shown that
linear-distributive quantales (i.e., a certain sort of
linear-distributive category) can be characterised by a variant of the
Frobenius condition, applied to the category Sup.
=====================================
Jonathon Funk
title: ``Purely Skeletal Geometric Morphisms''
Joint work with Marta Bunge.
abstract: We say that a monomorphism in a topos is pure if the constant
object 2 has the sheaf property with respect to every pullback of the
monomorphism. We say that a geometric morphism is purely skeletal if its
inverse image functor preserves pure monomorphisms. We discuss these
geometric morphisms in general, and in particular we discuss their role
in the theory of branched coverings in topos theory.
=====================================
Nicola Gambino
Title: Generalised species of structures.
Abstract: The notion of species of structures was introduced by Joyal in
the '80s to provide a categorical foundation for the use of formal power
series in combinatorics. Indeed, species of structures have a rich
calculus that corresponds to the one available for power series.
I will present a generalisation of the notion of species of structures
that allows one to give an abstract presentation of the calculus of
species and a homogenous account of the variants considered in the
literature. In particular, I will explain some of the 2-categorical
ideas necessary to organize generalised species in a cartesian closed
bicategory.
This is joint work with Marcelo Fiore, Martin Hyland and Glynn Winskel.
=====================================
Henrik Forssell
Algebraic Models of Intuitionist Theories
of Sets and Classes in text:
The paper constructs models of intuitionistic set theory in suitable
categories. First, a Basic Intuitionistic Set Theory (BIST) is
stated, and the categorical semantics are given. Second, we give a
notion of an ideal over a category, using which one can build
a model of BIST in which a given topos occurs as the sets. And
third, a sheaf model is given of a Basic Intuitionistic Class Theory
conservatively extending BIST. The paper extends the results of Awodey,
Butz, Simpson, and Streicher by
introducing a new and perhaps more natural notion of ideal, and in
the class theory of part three.
=====================================
Joachim Kock
Title: Weak units and realisation of homotopy 3-types
Abstract: First I'll outline a general approach to higher categories
with strict composition laws but only weak identity cells. The approach
is simplicial in spirit, but the base category is a certain category of
'coloured ordinals' instead of the usual Delta. Then I'll restrict
attention to the one-object case of dimension 3, and explain how strict
2-categories with a strict tensor product and weak units can realise
connected, simply connected homotopy 3-types. This example is the
first step towards Simpson's conjecture that every weak n-category is
equivalent to one with strict composition laws (but weak identity
cells).
=====================================
Eric Oliver Paquette
'Towards A Categorical Semantics For Topological Quantum Computing'
We show that C-colored manifolds (i.e. compact closed manifolds with
boundary where each boundary component is colored with an object of a
semisimple ribbon category) behaves in a similar manner as quantum
circuits under the action of a unitary modular functor. There, the set
of elementary gates is composed only of braid operations, rotations and
Dehn-twists.
We introduce the basic mathematical structure of a quantum circuit. We
then provide a complete development of a 2-dimensional CW-complex over
the markings of extended surfaces, where the later is connected and
simply-connected. Next, we provide a complete development of the
categorical framework in order to construct a C-extended unitary modular
functor (UMF) acting from the category of C-colored surfaces and
morphisms of C-colored surfaces to the category of finite-dimensional
vector spaces and linear isomorphisms.
From there, we conclude by giving a complete categorical semantics for
topological quantum computation including an abstract version of the
inner product, basic data units, basic data transformations, projectors
and the notion of topological invariance of the algorithms.
=====================================
Peter Selinger
Title: The CPM construction
Abstract: Abramsky and Coecke recently introduced the notion of a
"strongly compact closed category", which captures most of the notions
needed to model quantum mechanics. I will describe the graphical
calculus for these categories, and show how, from any strongly compact
closed category, one can construct another strongly compact closed
category of completely positive maps (the CPM construction). This
construction is relevant for the denotational semantics of quantum
programming languages and the modelling of quantum protocols.
=====================================
Benoit Valiron
Title : A Functional Programming Language
for Quantum Computation with Classical Control
Abstract :
The objective of this work is to develop a functional programming
language for quantum computation with classical control, following the
work of P. Selinger (2004) on quantum flow-charts. We construct a
lambda-calculus without side-effects to deal with quantum bits. We equip
this calculus with a probabilistic call-by-value operational semantics.
Since quantum information cannot be duplicated due to the {\it
no-cloning property}, we need a resource-sensitive type system. We
develop it based on affine intuitionistic linear logic. Unlike the
quantum lambda-calculus proposed by Van Tonder (2003, 2004), the
resulting lambda-calculus has only one lambda-abstraction, linear and
non-linear abstractions being encoded in the type system. We also
integrate classical and quantum data types within our language. The main
results of this work are the subject-reduction of the language and the
construction of a type inference algorithm.
=====================================
ALGEBRAIC SET THEORY: PREDICATIVITY
MICHAEL A. WARREN
There has been much interest of late in algebraic set theory as
introduced by Joyal and Moerdijk [2]. In this talk I will explain some
of the recent results which have been obtained in the algebraic study of
predicative set theories (cf. [1]).
REFERENCES
[1] Steve Awodey and Michael A. Warren, Predicative algebraic set
theory, Draft available at www.phil.cmu.edu/projects/ast/, 2004.
[2] A. Joyal and I. Moerdijk, Algebraic set theory, Cambridge University
Press, Cambridge, 1995.
=====================================
Mark Weber
"Braided and symmetric higher operads"
brief abstract: An aspect of the theory of higher (globular) operads, is
a precise understanding of globular pasting schemes, and these operads
are expressive enough to give a combinatorial definition of weak
n-category. It seems that the combinatorics of higher braidings and
symmetries appears implicitly, because for instance, a braided monoidal
category is a one object one arrow tricategory. One would like to
witness these higher symmetries more explicitly in the operadic
formulation. In this talk some variations on the existing higher operad
notion will be described, which mix the combinatorics of braids and
permutations in a natural way with that of globular pasting schemes.