The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.

Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.

Notions of ‘operad ’ and ‘multicategory ’ abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad ∗ on a category S, we define the term (S, ∗)-multicategory, subject to certain conditions on S and ∗. Different choices ofS and ∗ give some of the existing notions. We then describe the algebras for an (S, ∗)-multicategory and, finally, present a tentative selection of further developments. Our approach makes possible concise descriptions of Baez and Dolan’s opetopes and Batanin’s operads; both of these are included.

Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category theory, domain theory, logic, type theory, etc. In support of this proposal I will show how such an approach leads to interesting connections between various areas of computer science and mathematics; concentrating on one such example in some detail. Specifically, I will consider the line of my research involving denotational models of the pi calculus and algebraic theories with variable-binding operators, indicating how the abstract mathematical structure underlying these models fits with that of Joyal’s combinatorial species of structures. This analysis suggests both the unification and generalisation of models, and in the latter vein I will introduce generalised species of structures and their calculus. These generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc.) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.

In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law are precisely symmetric polycategories. 1

The purpose of this paper is to show how one may construct from a synchronous interaction category, such as SProc, a corresponding asynchronous version. Significantly, it is not a simple Kleisli construction, but rather arises due to particular properties of a monad combined with the existence of a certain type of distributive law. Following earlier work we consider those synchronous interaction categories which arise from model categories through a quotiented span construction: SProc arises in this way from labelled transition systems. The quotienting is determined by a cover system which expresses bisimulation. Asynchrony is introduced into a model category by a monad which, in the case of transition systems, adds the ability to idle. To form a process category atop this two further ingredients are required: pullbacks in the Kleisli category, and a cover system to express (weak) bisimulation. The technical results of the paper provide necessary and sufficient conditions for a Kleisli...

Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences

This paper presents a formal semantics for vectors and matrices, suitable for static type-checking. This is not available in apl, which produces run-time type errors, or in the usual functional languages, where matrices are typically implemented by lists of lists. Here, a matrix is a vector of vectors. Vectors are distinguished from lists by requiring that vector computations determine the length of the result from that of the argument, without reference to values. This leads to a two-level semantics, with values above and shapes below. Each operation must then specify its action on shapes as well as its action on values. Vectors and matrices inherit much of their structure from lists. In particular, the monadic structure given by singleton lists and the flattening of lists of lists extends in this way. Some new constructions, such as transposition of matrices, have no list counterpart. The power of this calculus for vector and matrix algebra is sufficient to represent the discrete Fou...

\Lambda LFCSUniv. of Edinburgh

Abstract. We develop a theory of double clubs which extends Kelly’s theory of clubs to the pseudo double categories of Paré and Grandis. We then show that the club for symmetric strict monoidal categories on Cat extends to a ‘double club ’ on the pseudo