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Definable additive categories: purity and model theory
About this Title
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:
2011; Volume 210, Number 987
ISBNs: 978-0-8218-4767-1 (print); 978-1-4704-0604-2 (online)
DOI: https://doi.org/10.1090/S0065-9266-2010-00593-3
Published electronically: July 27, 2010
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: Primary 03C60; Secondary 03C52, 16D90, 18C35, 18E05, 18E10, 18E35
Table of Contents
Chapters
- 1. Introduction
- 2. Preadditive and additive categories
- 3. Preadditive categories and their ind-completions
- 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. pp-Elimination of quantifiers
- 20. Ultraproducts
- 21. Pure-injectives 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
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.
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