Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory (Encyclopedia of Mathematics and its Applications, Series Number 97)

Researchers, teachers and graduate students in algebra and topology--familiar with the very basic notions of category theory--will welcome this categorical introduction to some of the key areas of modern mathematics, without being forced to study category theory. Rather, each of the eight largely independent chapters analyzes a particular subject, revealing the power and applicability of the categorical foundations in each case. The Book Offers Categorical Introductions To Order, Topology, Algebra, And Sheaf Theory, Suitable For Graduate Students, Teachers, And Researchers Of Pure Mathematics. Readers Familiar With The Most Basic Notions Of Category Theory Will Learn About The Main Tools That Are Used In Modern Categorical Mathematics But Are Not Readily Available In The Literature. Hence, In Eight Independent Chapters, The Reader Will Encounter Various Ways Of How To Study Spaces: Order-theoretically Via Their Open-set Lattices, As Objects Of A Fairly Abstract Category Via Their Interaction With Other Objects, Or Via Their Topoi Of Set-valued Sheaves. Likewise, Algebras Are Treated As Both Models For Lawvere's Algebraic Theories And Eilenberg-moore Algebras For Monads, But They Appear Also As The Objects Of An Abstract Category With Various Levels Of Exactness Conditions. The Abstract Methods Are Illustrated By Applications That, In Many Cases, Lead To Results Not Yet Found In More Traditional Presentations Of The Various Subjects, For Instance, On The Exponentiability Of Spaces And Embeddability Of Algebras. Suggestions For Further Studies And Research Are Also Given.--jacket. Ordered Sets Via Adjunction / R.j. Wood -- Locales / Jorge Picado, Aleš Pultr, And Anna Tozzi -- Functional Approach To General Topology / Maria Manuel Clementino, Eraldo Giuli, And Walter Tholen -- Regular, Protomodular, And Abelian Categories / Dominique Bourn And Marino Gran -- Aspects Of Monads / John Macdonald And Manuela Sobral -- Algebraic Categories / Maria Cristina Pedicchio And Fabrizio Rovatti -- Sheaf Theory / Claudia Centazzo And Enrico M. Vitale -- Beyond Barr Exactness / George Janelidze, Manuela Sobral, And Walter Tholen. Edited By Maria Cristina Pedicchio, Walter Tholen. Includes Bibliographical References And Index. The book gives a categorical introduction to some of the key areas of modern mathematics. Researchers, teachers and graduate students in algebra and topology familiar with the very basic notions of category theory will find all the advanced tools needed for their subjects, without being forced to study category theory for its own sake. 'Sets for Mathematics', by F.W. Lawvere and R. Rosebrugh, [5] is a ground-baking, undergraduate, set theory textbook.