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.