In this paper I compare two well studied approaches to topological semantics| the domain-theoretic approach, exempli ed by the category of countably based equilogical spaces, Equ, and Type Two E ectivity, exempli ed by the category of Baire space representations, Rep(B ). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is...

Our main aim is to present the connection between 2-calculus and Cartesian closed categories both in an untyped and purely syntactic setting. More specifically we establish a syntactic equivalence theorem between what we call categorical com-binatory logic and j-calculus with explicit products and projections, with fl and q-rules as well as with surjective pairing. "Combinatory logic" is of cou...

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 ...

Cartesian closed categories (CCC’s) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an ...

Berry's category of dI-domains with stable functions is a relatively intricate , yet elegant framework for semantics of programming languages. Despite over fteen years of work in the area, the exact reasons for dis-tributivity (Axiom d) and nitariness (Axiom I) have not been fully ex-plicated. This paper shows that Axiom d and Axiom I are important when one works within the realm of Scott-domai...

We show that a version of Martin-Löf type theory with extensional identity, a unit type N1,Σ,Π, and a base type is a free category with families (supporting these type formers) both in a 1and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We then show th...

In this paper we relate several objects from quite diverse areas of mathematics. Closed meanders are the configurations which arise when one or several disjoint closed Jordan curves in the plane intersect the horizontal axis transversely. The question of their connectivity also arises when evaluating traces in Temperley-Lieb algebras. The variant of open meanders is closely related to the detai...

An axis-parallel b-dimensional box is a Cartesian product R1 ×R2 × . . .× Rb where each Ri (for 1 ≤ i ≤ b) is a closed interval of the form [ai, bi] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1 × R2 × . . . × Rb,...

Cartesian closed categories (CCC's) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatiza-tion of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce and Longo leads to seven equalities. We show that the uniication problem for this theory is undecidable, thus settling an ...