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. 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 which express universal properties of sums, products, mapping sets, and natural number recursion. The distinctive features of Cantorian abstract sets, as contrasted with the variable and cohesive sets of geometry and analysis, are made explicit and taken as special axioms. Functor categories are introduced in order to model the variable sets used in geometry, and to illustrate the failure of the axiom of choice. An appendix provides an explicit introduction to necessary concepts from logic, and an extensive glossary provides a window to the mathematical landscape.

• The first undergraduate text that develops a foundation for set theory and logic based on categorical data
• Includes a complete mini-course in logic and an instructive glossary
• Exercises embedded in text to assist student in participating fully in the proofs and development

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.