The type-theoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that evaluation of type-expressions is independent from evaluation of programexpressions. We propose a new explanation based on \programming languages as indexed categories" and illustrates how ML can be extended to support higher order modules, by developing a category-theoretic semantics for a calculus of modules with dependent types. The paper outlines also a methodology, which may lead to a modular approach in the study of programming languages. Introduction The addition of module facilities to programming languages is motivated by the need to provide a better environment for the development and maintenance of large programs. Nowadays many programming languages include such facilities. Throughout the paper Standard ML (see [Mac85, HMM86, MTH90]) is taken as representative for these languages. The implementation of module facilities has been ...

This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists.