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.
In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
"...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