A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Therefore, it becomes important to organize data types systematically so that we can understand the relationship between one data type and another and investigate future directions which lead us to discover exciting new data types. There have been several approaches to systematically organize data types: algebraic specification methods using algebras, domain theory using complete partially ordered sets and type theory using the connection between logics and data types. Here, we use category theory. Category theory has proved to be remarkably good at revealing the nature of mathematical objects, and we use it to understand the true nature of data types in programming.

To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is first-order acyclic sharing graphs represented by let-syntax, and others are extensions with higher-order constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in ’ to the free construction.

The generality of a derivation is an equivalence relation on the set of occurrences of variables in its premises and conclusion such that two occurrences of the same variable are in this relation if and only if they must remain occurrences of the same variable in every generalization of the derivation. The variables in question are propositional or of another type. A generalization of the derivation consists in diversifying variables without changing the rules of inference. This paper examines in the setting of categorial proof theory the conjecture that two derivations with the same premises and conclusions stand for the same proof if and only if they have the same generality. For that purpose generality is defined within a category of equivalence relations on finite ordinals, where composition is rather complicated. Several examples are given of deductive systems of derivations covering fragments of

What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...

It is shown that the multiplicative monoids of Temperley-Lieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. Thereby one obtains a representation of braid groups in matrices, which, though different and presumably new, is related to the standard representation of braid groups in Temperley-Lieb algebras. Mathematics Subject Classification (2000): 57M99, 20F36, 18A40 1

This paper gives a simple presentation of the free star-autonomous category over a category, based on Eilenberg-Kelly-MacLane graphs and Trimble rewiring, for full coherence.

An action : V A! A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V ! [A; A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V ! [A; A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functor|so that, passing to the categories of monoids (also called \algebras") in V and in [A; A], we have an adjunction MonF a MonG between the category MonV of monoids in V and the category Mon[A; A] = MndA of monads on A. We give sucient conditions for the existence of the right adjoint g, which involve the existence of right adjoints for the functors X { and { A, and make A (at least when V is symmetric and closed) into a tensored and cotensored V-category A. We give explicit formulae, as large ends, for the right adjoints g and MonG, and also for some related right adjoints, when they exist; as well as another explicit expression for MonG as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.

We introduce an abstract notion of POVM within the categorical quantum mechanical semantics in terms of