**Memoirs of the American Mathematical Society**

2011;
109 pp;
Softcover

MSC: Primary 03;
Secondary 16; 18

Print ISBN: 978-0-8218-4767-1

Product Code: MEMO/210/987

List Price: $74.00

Individual Member Price: $44.40

**Electronic ISBN: 978-1-4704-0604-2
Product Code: MEMO/210/987.E**

List Price: $74.00

Individual Member Price: $44.40

# Definable Additive Categories: Purity and Model Theory

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*Mike Prest*

Most of the model theory of modules works, with only minor modifications, in much more general additive contexts (such as functor categories, categories of comodules, categories of sheaves). Furthermore, even within a given category of modules, many subcategories form a “self-sufficient” context in which the model theory may be developed without reference to the larger category of modules. The notion of a definable additive category covers all these contexts. The (imaginaries) language which one uses for model theory in a definable additive category can be obtained from the category (of structures and homomorphisms) itself, namely, as the category of those functors to the category of abelian groups which commute with products and direct limits. Dually, the objects of the definable category—the modules (or functors, or comodules, or sheaves)—to which that model theory applies may be recovered as the exact functors from the, small abelian, category (the category of pp-imaginaries) which underlies that language.

#### Table of Contents

# Table of Contents

## Definable Additive Categories: Purity and Model Theory

- Chapter 1. Introduction 18 free
- Chapter 2. Preadditive and additive categories 512 free
- Chapter 3. Preadditive categories and their ind-completions 916
- Chapter 4. The free abelian category of a preadditive category 1522
- Chapter 5. Purity 2128
- Chapter 6. Locally coherent categories 2936
- Chapter 7. Localisation 3138
- Chapter 8. Serre subcategories of the functor category 3340
- Chapter 9. Conjugate and dual categories 3744
- Chapter 10. Definable subcategories 3946
- Chapter 11. Exactly definable categories 4350
- Chapter 12. Recovering the definable structure 4754
- Chapter 13. Functors between definable categories 5360
- Chapter 14. Spectra of definable categories 5764
- Chapter 15. Definable functors and spectra 5966
- Chapter 16. Triangulated categories 6370
- Chapter 17. Some open questions 6572
- Chapter 18. Model theory in finitely accessible categories 6774
- Chapter 19. pp-Elimination of quantifiers 7380
- Chapter 20. Ultraproducts 7582
- Chapter 21. Pure-injectives and elementary equivalence 7784
- Chapter 22. Imaginaries and finitely presented functors 8390
- Chapter 23. Elementary duality 8794
- Chapter 24. Hulls of types and irreducible types 8996
- Chapter 25. Interpretation functors 9198
- Chapter 26. Stability 99106
- Chapter 27. Ranks 101108
- Bibliography 105112