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 [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 mat...

We extend the Theory of Computation on real numbers, continuous real functions, and bounded closed Euclidean subsets, to compact metric spaces (X, d): thereby generically including computational and optimization problems over higher types, such as the compact ‘hyper’ spaces of (i) nonempty closed subsets of X w.r.t. Hausdorff metric, and of (ii) equicontinuous functions on X . The thus obtained...

For a small category K enriched over a suitable monoidal category V , the free completion of K under colimits is the presheaf category [K ,V ]. If K is large, its free completion under colimits is the V -category PK of small presheaves on K , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed struct...

This paper describes a category theory semantics for conceptual data modeling. The conceptual data modeling technique used can be seen as a generalization of most existing conceptual data modeling techniques. It contains features such as specialization, generalization , and power types. The semantics uses only simple category theory constructs such as (co)limits and epi-and monomorphisms. There...

A ‘Chen space’ is a set X equipped with a collection of ‘plots’ — maps from convex sets to X — satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth ma...

We show that a certain simple call-by-name continuation semantics of Parigot’s λμ-calculus is complete. More precisely, for every λμ-theory we construct a cartesian closed category such that the ensuing continuation-style interpretation of λμ, which maps terms to functions sending abstract continuations to responses, is full and faithful. Thus, any λμ-category in the sense of L. Ong (1996, in “...

We present a generalization of cartesian closed categories (CCCs) for dependent types, called dependent cartesian closed categories (DCCCs), which also provides a reformulation of categories with families (CwFs), an abstract semantics for Martin-Löf type theory (MLTT) which is very close to the syntax. Thus, DCCCs accomplish mathematical elegance as well as a direct interpretation of the syntax...

We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint universal binary coproducts.

There is a very close and well-understood relationship between proofs in intuitionistic logic, simply typed lambda-terms, and morphisms in Cartesian closed categories. The same relationship can be established for multiplicative linear logic (MLL), where proof nets take the role of the lambda-terms, and starautonomous categories the role of Cartesian closed categories. It is certainly desirable ...

We give two classes of sound and complete models for the computational-calculus, or c-calculus. For the rst, we generalise the notion of cartesian closed category to that of closed Freyd-category. For the second, we generalise simple indexed categories. The former gives a direct semantics for the computational-calculus. The latter corresponds to an idealisation of stack-based intermediate langu...