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General concepts and methods that occur throughout mathematics & ndash; and now also in theoretical computer science & ndash; are the subject of this book. It is a thorough introduction to Categories, emphasizing the geometric nature of the subject and explaining its connections to mathematical logic. The book should appeal to the inquisitive reader who has seen some basic topology and algebra and would like to learn and explore further. The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint functors, and topoi. A special feature of the work is a general calculus of relations presented in the second part. This calculus offers another, often more amenable framework for concepts and methods discussed in part one. Some aspects of this approach find their origin in the relational calculi of Peirce and Schroeder from the last century, and in the 1940's in the work of Tarski and others on relational algebras. The representation theorems discussed are an original feature of this approach.

Category theory. Homological algebra

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Commutative algebra. --- Homology theory. --- 512.66 --- 512.66 Homological algebra --- Homological algebra --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Algebra --- Commutative algebra --- Homology theory

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Category theory. Homological algebra --- 512.736 --- Serre cohomology. K-theory --- Hodge theory. --- 512.736 Serre cohomology. K-theory --- 51 --- 51 Mathematics --- Mathematics --- Geometry, Algebraic. --- K-theory.

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This is an extended treatment of the set-theoretic techniques which have transformed the study of abelian group and module theory over the last 15 years. Part of the book is new work which does not appear elsewhere in any form. In addition, a large body of material which has appeared previously (in scattered and sometimes inaccessible journal articles) has been extensively reworked and in many cases given new and improved proofs. The set theory required is carefully developed with algebraists in mind, and the independence results are derived from explicitly stated axioms. The book contains exe

Abelian groups --- Modules (Algebra) --- Set theory --- #KVIV:BB --- 512.66 --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Logic, Symbolic and mathematical --- Mathematics --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Commutative groups --- Group theory --- 512.66 Homological algebra --- Homological algebra --- Abelian groups. --- Modules (Algebra). --- Set theory. --- Physical Sciences & Mathematics

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General concepts and methods that occur throughout mathematics - and now also in theoretical computer science - are the subject of this book. It is a thorough introduction to Categories, emphasizing the geometric nature of the subject and explaining its connections to mathematical logic. The book should appeal to the inquisitive reader who has seen some basic topology and algebra and would like to learn and explore further.The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint fun

Category theory. Homological algebra --- Allegories (Mathematics). --- Categories (Mathematics). --- Categories (Mathematics) --- Allegories (Mathematics) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Toposes. --- Foncteurs, Théorie des --- Topos (mathématiques) --- Faisceaux