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# memo_has_moved_text(); Definable additive categories: purity and model theory

Mike Prest, School of Mathematics, Alan Turing Building, University of Manchester, Manchester M13 9PL, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year 2010: Volume 210, Number 987
ISBNs: 978-0-8218-4767-1 (print); 978-1-4704-0604-2 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2010-00593-3
Published electronically: July 27, 2010
MathSciNet review: 2791358
Keywords: additive category, abelian category, definable category, module, functor category, finitely accessible, locally coherent, pure-injective, free abelian category, pp-formula, pp-type, ultraproduct, Serre subcategory, interpretation, imaginary, elementary duality
MSC (2010): Primary 03C60; Secondary 03C52, 16D90, 18C35, 18E05, 18E10, 18E35

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Chapters

• Chapter 1. Introduction
• Chapter 3. Preadditive categories and their ind-completions
• Chapter 4. The free abelian category of a preadditive category
• Chapter 5. Purity
• Chapter 6. Locally coherent categories
• Chapter 7. Localisation
• Chapter 8. Serre subcategories of the functor category
• Chapter 9. Conjugate and dual categories
• Chapter 10. Definable subcategories
• Chapter 11. Exactly definable categories
• Chapter 12. Recovering the definable structure
• Chapter 13. Functors between definable categories
• Chapter 14. Spectra of definable categories
• Chapter 15. Definable functors and spectra
• Chapter 16. Triangulated categories
• Chapter 17. Some open questions
• Chapter 18. Model theory in finitely accessible categories
• Chapter 19. pp-Elimination of quantifiers
• Chapter 20. Ultraproducts
• Chapter 21. Pure-injectives and elementary equivalence
• Chapter 22. Imaginaries and finitely presented functors
• Chapter 23. Elementary duality
• Chapter 24. Hulls of types and irreducible types
• Chapter 25. Interpretation functors
• Chapter 26. Stability
• Chapter 27. Ranks

### Abstract

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. The additive functors between definable categories which commute with products and direct limits are precisely the functors given by interpretations (using pp formulas); they are in natural duality with the exact functors between the corresponding categories of pp-imaginaries. All this, including relevant background on (pre)additive categories, is presented and is followed by a development of various aspects of model theory in definable additive categories.