Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

This first textbook on formal concept analysis gives a systematic presentation of the mathematical foundations and their relations to applications in computer science, especially in data analysis and knowledge processing. Above all, it presents graphical methods for representing conceptual systems that have proved themselves in communicating knowledge. The mathematical foundations are treated thoroughly and are illuminated by means of numerous examples, making the basic theory readily accessible in compact form.

The use of highly abstract mathematical frameworks is essential for building the sort of theoretical foundation for semantic integration needed to bring it to the level of a genuine engineering discipline. At the same time, much of the work that has been done by means of these frameworks assumes a certain amount of background knowledge in mathematics that a lot of people working in ontology, even at a fairly high theoretical level, lack. The major purpose of this short paper is (...) provide a (comparatively) simple model of semantic integration that remains within the friendlier confines of first-order languages and their usual classical semantics and logic. (shrink)

Ontology today is in many ways in a state similar to that of analysis in the late 18th century prior to arithmetization: it lacks the sort rigorous theoretical foundations needed to elevate ontology to the level of a genuine scientific discipline. This paper attempts to make some first steps toward the development of such foundations. Specifically, starting with some basic intuitions about ontologies and their content, I develop an expressively rich framework capable of treating ontologies as theoretical objects whose properties (...) and logical inter- connections — notably, potential for integration — we can clearly define and study. (shrink)

This paper gives a general definition of a “kind of schema” along with general definitions for the schemas of a species, and for the databases, constraints, and queries over a given schema of a species. This leads naturally to a general theory of data translation and integration over arbitrary schemas of arbitrary species, based on schema morphisms, and to a similar general theory of ontology translation and integration over arbitrary logics. Institutions provide a general notion of logic, and Grothendieck flattening (...) provides a general tool for integrating heterogeneous schemas, species and logics, as well as theories, such as ontologies, over different logics. Many examples of our novel concepts are included, some rather detailed. An initial section introduces data integration and ontologies for readers who are not specialists, with some emphasis on challenges. A brief review of universal algebra is also given, though some familiarity with category theory is assumed in later sections. (shrink)