eBook ISBN:  9781470406042 
Product Code:  MEMO/210/987.E 
List Price:  $74.00 
MAA Member Price:  $66.60 
AMS Member Price:  $44.40 
eBook ISBN:  9781470406042 
Product Code:  MEMO/210/987.E 
List Price:  $74.00 
MAA Member Price:  $66.60 
AMS Member Price:  $44.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 210; 2011; 109 ppMSC: Primary 03; Secondary 16; 18
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 “selfsufficient” 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 ppimaginaries) which underlies that language.

Table of Contents

Chapters

1. Introduction

2. Preadditive and additive categories

3. Preadditive categories and their indcompletions

4. The free abelian category of a preadditive category

5. Purity

6. Locally coherent categories

7. Localisation

8. Serre subcategories of the functor category

9. Conjugate and dual categories

10. Definable subcategories

11. Exactly definable categories

12. Recovering the definable structure

13. Functors between definable categories

14. Spectra of definable categories

15. Definable functors and spectra

16. Triangulated categories

17. Some open questions

18. Model theory in finitely accessible categories

19. ppElimination of quantifiers

20. Ultraproducts

21. Pureinjectives and elementary equivalence

22. Imaginaries and finitely presented functors

23. Elementary duality

24. Hulls of types and irreducible types

25. Interpretation functors

26. Stability

27. Ranks


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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 “selfsufficient” 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 ppimaginaries) which underlies that language.

Chapters

1. Introduction

2. Preadditive and additive categories

3. Preadditive categories and their indcompletions

4. The free abelian category of a preadditive category

5. Purity

6. Locally coherent categories

7. Localisation

8. Serre subcategories of the functor category

9. Conjugate and dual categories

10. Definable subcategories

11. Exactly definable categories

12. Recovering the definable structure

13. Functors between definable categories

14. Spectra of definable categories

15. Definable functors and spectra

16. Triangulated categories

17. Some open questions

18. Model theory in finitely accessible categories

19. ppElimination of quantifiers

20. Ultraproducts

21. Pureinjectives and elementary equivalence

22. Imaginaries and finitely presented functors

23. Elementary duality

24. Hulls of types and irreducible types

25. Interpretation functors

26. Stability

27. Ranks