Types 83 9.1 Syntax : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 9.2 Typing Rules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83 9.3 Operational Semantics : : : : : : : : : : : : : : : : : : : : : : : : 84 iv 9.4 Impredicative Existentials : : : : : : : : : : : : : : : : : : : : : : 84 9.5 Representation Independence : : : : : : : : : : : : : : : : : : : : 85 9.6 Projection Notation : : : : : : : : : : : : : : : : : : : : : : : : : 85 9.7 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 10 Modularity 88 10.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 88 10.2 A Critique of Some Modularity Mechanisms : : : : : : : : : : : : 88 10.3 Basic Modules : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 10.4 Module Hierarchies : : : : : : : : : : : : : : : : : : : : : : : : : : 97 10.5 Parameterized Modules : : : : : : : : : : : : : : : : : : : : : : : 98 10.6 References : : : : : : : : : : : ...

Abstract. This paper gives an overview of XML formal models, summarizes database engineering practices, problems and their evolution. We focus on categorical aspects of XML formal models. Many formal models such as XML Data Model, XQuery Data Model or Algebra for XML can be described in terms of category theory. This kind of description allows to consider generic properties of these formalisms, e.g. expressive power, optimization, reduction or translation between them, among others. These properties are rather crucial to comparison of different XML formal models and to consequent decision which formal system should be used to solve a concrete problem. This work aim is to be the basis for further research in the area of XML formal models where category theory is applied. 1

We present a fibrational semantics of positive Horn logic with and without equality. The corresponding soundness, completeness and classification theorems are established. Emphasis lies on the results as well as on the proof methods.

This work expounds the notion that (structured) categories are syntax free presentations of type theories, and shows some of the ideas involved in deriving categorical semantics for given type theories. It is intended for someone who has some knowledge of category theory and type theory, but who does not fully understand some of the intimate connections between the two topics. We begin by showing how the concept of a category can be derived from some simple and primitive mechanisms of monadic type theory. We then show how the notion of a category with finite products can model the most fundamental syntactical constructions of (algebraic) type theory. The idea of naturality is shown to capture, in a syntax free manner, the notion of substitution, and therefore provides a syntax free coding of a multiplicity of type theoretical constructs. Using these ideas we give a direct derivation of a cartesian closed category as a very general model of simply typed λ-calculus with binary products and a unit type. This article provides a new presentation of some old ideas. It is intended to be a tutorial paper aimed at audiences interested in elementary categorical type theory. Further details can be found in [Cro93]. 1 1