Category theory offers a unified mathematical framework for the study of specifications and programs in a variety of styles, such as procedural, functional and concurrent. One way that these different languages may be treated uniformly is by generalising the definitions of some standard categorical concepts. In this paper we reproduce in the generalised theory analogues of some standard theorems on isomorphism, and outline their applications to programming languages.

The subject of linear logic has recently become very important in theoretical computer science. It is apparent that the *-autonomous categories studied at length in by Barr (1979) are a model for a large fragment of linear logic, although not quite for the whole thing. Since the main reference is out of print and since large parts of that volume are devoted to results highly peripheral to the matter at hand, it seemed reasonable to provide a short introduction to the subject.

We combine the principles of the Floyd-Warshall-Kleene algorithm, enriched categories, and Birkhoff arithmetic, to yield a useful class of algebras of transitive vertex-labeled spaces. The motivating application is a uniform theory of abstract or parametrized time in which to any given notion of time there corresponds an algebra of concurrent behaviors and their operations, always the same operations but interpreted automatically and appropriately for that notion of time. An interesting side application is a language for succinctly naming a wide range of datatypes.

Various Theories of Types are introduced, by stressing the analogy ‘propositions-as-types’: from propositional to higher order types (and Logic). In accordance with this, proofs are described as terms of various calculi, in particular of polymorphic (second order) λ-calculus. A semantic explanation is then given by interpreting individual types and the collection of all types in two simple categories built out of the natural numbers (the modest sets and the universe of ω-sets). The first part of this paper (syntax) may be viewed as a short tutorial with a constructive understanding of the deduction theorem and some work on the expressive power of first and second order quantification. Also in the second part (semantics, §§6–7) the presentation is meant to be elementary, even though we introduce some new facts on types as quotient sets in order to interpret ‘explicit polymorphism’. (The experienced reader in Type Theory may directly go, at first reading, to §§6–8).