Synopses & Reviews
Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion.
Review
"To learn set theory this way means not having to relearn it later.... Recommended." Choice
Review
"...the categorical approach to mathematics has never been presented with greater conviction than it has in this book. The authors show that the use of categories in analyzing the set concept is not only natural, but inevitable." Mathematical Reviews"To learn set theory this way means not having to relearn it later.... Recommended." Choice
Synopsis
This volume has arisen from courses given by the authors to students at the State University of New York at Buffalo. The aim is to give an elementary view of set theory which will be useful to readers without a substantial background in math. The approach taken by the authors is grounded in their own interests in category theory. This gives the book a character that is different from other books at a similar level and will make it of interest to readers already acquainted with set theory.
Synopsis
Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time in a text, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. An Appendix provides an explicit introduction to necessary concepts from logic, and an extensive Glossary provides a window to the mathematical landscape.
Synopsis
For the first time in a textbook, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
Table of Contents
Foreword; 1. Abstract sets and mappings; 2. Sums, monomorphisms and parts; 3. Finite inverse limits; 4. Colimits, epimorphisms and the axiom of choice; 5. Mapping sets and exponentials; 6. Summary of the axioms and an example of variable sets; 7. Consequences and uses of exponentials; 8. More on power sets; 9. Introduction to variable sets; 10. Models of additional variation; Appendices; Bibliography.