Elementary Categories, Elementary Toposes [Paperback]

The book covers elementary aspects of category theory and topos theory for graduate students in mathematics, computer science, and logic; it has few mathematical prerequisites, and uses categorical methods throughout, rather than beginning with set theoretical foundations. Working with key concepts such as Cartesian closedness, adjunctions, regular categories, and the internal logic of a topos, the book features full statements and elementary proofs for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Other chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis.

PART I: Categories
1. Rudimentary structures in a Category
2. Products, Equalizers, and their Duals
3. Groups
4. Sub-Objects, Pullbacks, and Limits
5. Relations
6. Cartesian Closed Categories
7. Product Operators and Others
PART II: The Category of Categories
8. Functors and Categories
9. Natural Transformations
10. Adjunctions
11. Slice Categories
12. Mathematical Foundations
PART III: Toposes
13. Basics
14. The Internal Language
15. A Soundness Proof for Topos Logic
16. From the Internal Language to the Topos
17. The Fundamental Theorem
18. External Semantics
19. Natural Number Objects
20. Categories in a Topos
21. Topologies
PART IV: Some Toposes
22. Sets
23. Synthetic Differential Geometry
24. The Effective Topos
25. Relations in Regular Categories

An exceptionally clearly written and wide-ranging introduction to category and topos theory. . . . packed with things interesting to the expert, yet presented in a manner intelligible to the beginner. . . . contains a wealth of thought-provoking exercises. --Journal of Symbolic Logic

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