Interpreting j-conversion as an expansion rule in the simply-typed -calculus maintains the confluence of reduction in a richer type structure. This use of expansions is supported by categorical models of reduction, where fi-contraction, as the local counit, and j-expansion, as the local unit, are linked by local triangle laws. The latter form reduction loops, but strong normalisation (to the long fij-normal forms) can be recovered by "cutting" the loops.

The notion of an Institution [5] is here taken as the precise formulation for the notion of a logical system. By using elementary tools from the core of category theory, we are able to reveal the underlying mathematical structures lying "behind" the logical formulation of the satisfaction condition, and hence to acquire a both suitable and deeper understanding of the institution concept. This allows us to systematically approach the problem of describing and analyzing relations between logical systems. Theorem 2.10 redesigns the notion of an institution to a purely categorical level, so that the satisfaction condition becomes a functorial (and natural) transformation from specifications to (subcategories of) models and vice versa. This systematic procedure is also applied to discuss and give a natural description for the notions of institution morphism and institution map. The last technical discussion is a careful and detailed analysis of two examples, which tries to outl...

ions are defined both for objects and layers. There are several compatibility requirements for the definition of these functions. The set of objects contains a specific ?-element which allows the source and target functions to be total on the set of objects. Up to now there exists no implementation of general colimits in the AGG-system. This problem is currently fixed by the integration of the colimit library. Again we can use the colimit computation for Alpha algebras. For this purpose we have to find an Alpha representation of AGG-graphs. Here we will outline the idea. r0 r1 r2 object layer label v0 r4 r0 Item Data The picture above presents a possible Alpha type algebra for AGG-graphs. r 0 ; r 1 and r 2 correspond to the abstraction, source and target functions, r 4 represents the assignment of layers to objects and v 0 is the labelling function. Note that although not shown in the picture, since all references are total, r 1 ; r 2 and r 3 are defined also for layer. This shows ...

Machine by Curien [Cur86]. It is based upon a weak categorical combinatory logic, viz. lacking surjective pairing and extensionality, that arose as a direct semantic-to-syntactic translation of the lambda calculus of tuples. The computational mode was combinator term reduction through rewriting using a direct left-to-right parse algorithm, initially making the evaluation strategy inefficiently eager 1 . Application is therefore simply juxtaposition, losing the full expressiveness of-reduction that computes via substitution. Its overly strong bias towards the lambda calculus was another factor that limited its expressiveness. On one hand the CAM demanded the existence of categorical products but on the other it had no coproducts for developing many useful data structures. Nevertheless, the high acceptance and efficiency of the CAM-based ML compiler, CAML, gives significant encouragement towards developing a highly-programmable categorical computing paradigm. Some prominent workers in ...

. With the notions of partial morphism and relation to be understood with respect to a class M of monomorphisms in a finitely complete category C, we give sufficient conditions for the graph functor Par(C) ! Rel(C) to admit a right adjoint. Only under an additional condition is this right adjoint given by the naturally constructed "pierced power objects". Key words: partial map, relation, pierced power object, element relation Subject classification: 18B10, 18B25, 18A30 1. Introduction Mostly, but not exclusively, motivated by the interest theoretical computer scientists have in them, in recent years many authors have considered relations from a categorical point of view (see, for example, [CKW91], [Don96], [Fio95], [Jay90], [Jay91], [Pav95], [RR88], [Wyl91]). The global problem of presenting the assignment C 7! Rel(C) as a "good functor" has been a prominent theme in many of these works. In this paper we revert back to the local problem of examining the embedding C ! Rel(C), more ...