Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis, Durham, July 9-21, 1977 (Lecture Notes in Mathematics (753))

It is the task of history to explain how modern concepts arose and why they are interesting and relevant.It is the task of modern mathematics to explain the same thing independent of and irrespective of history. Sheaf theory, not really being a subject, cannot properly be said to have a history. Rather, it is an octopus spreading itself throughout everyone else's history. Of course, "everyone" is an exaggeration since sheaf theory is a part of geometry; namely, that part concerned with the passage from local properties to global properties. For instance, in complex analysis there is a sharp distinction between studying convergent power series at a point and investigating the Riemann surface of a whole "analytic configuration". Similarly, in algebraic topology, chain complexes with supports in a topological space have similar aspects, but this dichotomy was not so evident until Cartan clarified it and provided the major toolcohomology with coefficients in a sheaf -which ever since has mediated the passage from local to global. The description of a sheaf as an etale space (due to Lazard), developed by Cartan in 1950 , was and is adequate for nearly all of the geometric uses of sheaf theory. Analytic geometry, differential geometry, and differential equations are ]E 93 Fragments of the history of sheaf theory....Pages 1-79 Finiteness and decidability:I....Pages 80-100 Injective banach sheaves....Pages 101-112 Simplicial sets and the foundations of analysis....Pages 113-124 Localization with respect to a measure....Pages 125-156 On the concept of a measurable space I....Pages 157-168 Banach spaces in categories of sheaves....Pages 169-196 The affine scheme of a general ring....Pages 197-211 Localisation, spectra and sheaf representation....Pages 212-238 Concrete quasitopoi....Pages 239-254 Higher dimensional torsors and the cohomology of topoi : The abelian theory....Pages 255-279 Sheaf models for analysis....Pages 280-301 Sheaves and logic....Pages 302-401 Heyting-valued models for intuitionistic set theory....Pages 402-414 Sheaf theoretical concepts in analysis: Bundles and sheaves of Banach spaces, Banach C(X)-modules....Pages 415-441 Continuity in spatial toposes....Pages 442-465 A syntactic approach to Diers' localizable categories....Pages 466-478 Conditions related to de Morgan's law....Pages 479-491 Sheaves in physics — Twistor theory....Pages 492-499 Sheaf representations and the dedekind reals....Pages 500-513 Manifolds in formal differential geometry....Pages 514-533 Note on non-abelian cohomology....Pages 534-541 Representations of rings and modules....Pages 542-585 Cramer's rule in the Zariski topos....Pages 586-594 On the spectrum of a real representable ring....Pages 595-611 On functorializing usual first-order model theory....Pages 612-622 Topos theory and complex analysis....Pages 623-659 Identity and existence in intuitionistic logic....Pages 660-696 Weak adjointness in proof theory....Pages 697-701 Rank one projective modules over certain fourier algebras....Pages 702-713 Boolean valued analysis....Pages 714-731 Sheaf-theoretical methods in the solution of Kaplansky's problem....Pages 732-738 Generic Galois theory of local rings....Pages 739-767 Sheaf theory and zero-dimensional mappings....Pages 768-779 "The database industry is clearly a multi-billion, world-wide, all-encompassing part of the software world. This is part thanks to the standardization of query languages in the form of SQL. And yet, it is well known that SQL has significant shortcomings. Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages analyzes one of those shortcomings, the way in which quantification is dealt with in SQL. It is well known that most query languages are simply versions of First Order Logic (FOL). GQs are an extension of the idea of quantifier in FOL. Hence, GQs can be a meaningful extension of the treatment of quantification in query languages. Even though studied within the theoretical community up until now, GQs can be successfully applied, and are a perfect example of a practical theory within databases. This book provides a brief background in logic and introduces the concept of GQs, and then develops a query language based on GQs, called QLGQ. Using QLGQ, the reader explores the efficient implementation of the concept, always a primary consideration in databases. This professional book also includes several extensions, for use with documents employing question and answer techniques. Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages is the result of several years of research funded by NSF through a CAREER Award. It is designed for practitioners and researchers that work within the database management field. This volume is also suitable for graduate-level students in computer science."--Publisher's website The database industry is a multi-billion, world-wide, all-encompassing part of the software world. Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages introduces a query language called GQs—Generalized Quantification in Query. Most query languages are simply versions of First Order Logic (FOL). GQs are an extension of the idea of quantifier in FOL. GQs are a perfect example of a practical theory within databases. This book provides a brief background in logic and introduces the concept of GQs, and then develops a query language based on GQs. Using Query Language with Generalized Quantifiers, the reader explores the efficient implementation of the concept, always a primary consideration in databases. This professional book also includes several extensions for use with documents employing question and answer techniques. Designed for practitioners and researchers within the database management field; also suitable for advanced-level students in computer science.