We exhibit confluent and effectively weakly normalizing (thus decidable) rewriting systems for the full equational theory underlying cartesian closed categories, and for polymorphic extensions of it. The λ-calculus extended with surjective pairing has been well-studied in the last two decades. It is not confluent in the untyped case, and confluent in the typed case. But to the best of our knowl...

We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to deene a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory.

We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Köthe sequence spaces. In this setting, the “of course” connective of linear logic has a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. ...

The category D of finite directed graphs is cartesian closed, hence it has a product and exponential objects. For a fixed K, let K be the class of all directed graphs of the formK, preordered by the existence of homomorphisms, and quotiented by homomorphic equivalence. It has loong been known that K, is always boolean lattice. In this paper we prove that for any complete graph Kn with n ≥ 3, K ...

In [Wells, 1990] the second author introduced the notion of form, a graph-based method of specification of mathematical structures that generalizes Ehresmann’s sketches. In [Bagchi and Wells, 1996], the authors produced a structure for forms which provides a uniform proof theory based on finite-limit constructions for many types of forms, including all types of sketches and also forms that can ...

We use continuity spaces, a common refinement of posets and metric spaces, to develop a general theory of semantic domains which includes metric spaces and domains of cpo’s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming. Our proposal for a general notion ...

We introduce the notion of cartesian closed double category to provide mobile calculi for communicating systems with specific semantic models: One dimension is dedicated to compose systems and the other to compose their computations and their observations. Also, inspired by the connection between simply typed -calculus and cartesian closed categories, we define a new typed framework, called dou...

Countably based filter spaces have been suggested in the 1970’s as a model for recursion theory on higher types. Weak limit spaces with a countable base are known to be the class of spaces which can be handled by the Type-2 Model of Effectivity (TTE). We prove that the category of countably based proper filter spaces is equivalent to the category of countably based weak limit spaces. This resul...

The category of L-domains was discovered by A. Jung while solving the problem of nding maximal cartesian closed categories of algebraic CPO's and continuous functions. In this note we analyse properties of the lossless powerdo-main construction, that is closed on the algebraic L-domains. The powerdomain is shown to be isomorphic to a collection of subsets of the domain on which the construction...

In our paper [1] we have introduced the basic concepts and facts for scientific problem solving by help of mathematical machines, i.e. by logical reasoning about programming of these machines. These fundamental concepts were: category, cartesian closed category, diagram and limit, topos and elementary topos, but the most important was the concept of basic types. Basic types actually form the st...