The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higher-order processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higher-order processes. For this it is useful to generalise event structures to allow events which “persist.”

Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the higher rank graph for the associated C ∗ –algebra to be: simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from “commuting” rank 1 graphs is given.

Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the A-equivariant L-function of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order A in A for which there exists a ‘projective A-structure ’ on M. The existence of such a structure is guaranteed if A is a maximal order, and also occurs in many natural examples where A is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.

Abstract. We study a generalization of the Fourier-Mukai transform for smooth projective varieties. We find conditions under which the transform satisfies an inversion theorem. This is done by considering a series of four conditions on such transforms which increasingly constrain them. We show that a necessary condition for the existence of such transforms is that the first Chern classes must vanish and the dimensions of the varieties must be equal. We introduce the notion of bi-universal sheaves. Some examples are discussed and new applications are given, for example, to prove that on polarised abelian varieties, each Hilbert scheme of points arises as a component of the moduli space of simple bundles. The transforms are used to prove the existence of numerical constraints on the Chern classes of stable bundles.

... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various side-conditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the r-cube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the

Non-commutative projective bundle

Let L be a 1-reduced simplicial set. Let G(−) and C(−) be the Kan loop group and normalized chain functors respectively. The explicit, natural twisting cochain tL: CL → C(GL) of Szczarba [16] determines a natural morphism of chain algebras θL: ΩCL → C(GL) that induces an isomorphism in homology, since CL

Abstract. Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids.

Summary. We prove that iteration theories can be introduced as algebras for the monad Rat on the category of signatures assigning to every signature Σ the rational-Σ-tree signature. This supports the result that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: Rat is the monad of free rational theories and every free rational theory has a continuous completion. Key words: Iteration theory, rational theory, monad, Eilenberg-Moore algebra. 1

Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning about the modularity of specific properties for specific ways of combining specific forms of rewriting. This paper is, we believe, the first to ask a much more general question. Namely, what can be said about modularity independently of the specific form of rewriting, combination and property at hand. A priori there is no reason to believe that anything can actually be said about modularity without reference to the specifics of the particular systems etc. However, this paper shows that, quite surprisingly, much can indeed be said.