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Definable Additive Categories: Purity and Model Theory
Mike Prest, University of Manchester, England

Memoirs of the American Mathematical Society
2011; 109 pp; softcover
Volume: 210
ISBN-10: 0-8218-4767-8
ISBN-13: 978-0-8218-4767-1
List Price: US$74
Individual Members: US$44.40
Institutional Members: US$59.20
Order Code: MEMO/210/987
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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 "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

  • Introduction
  • Preadditive and additive categories
  • Preadditive categories and their ind-completions
  • The free abelian category of a preadditive category
  • Purity
  • Locally coherent categories
  • Localisation
  • Serre subcategories of the functor category
  • Conjugate and dual categories
  • Definable subcategories
  • Exactly definable categories
  • Recovering the definable structure
  • Functors between definable categories
  • Spectra of definable categories
  • Definable functors and spectra
  • Triangulated categories
  • Some open questions
  • Model theory in finitely accessible categories
  • pp-Elimination of quantifiers
  • Ultraproducts
  • Pure-injectives and elementary equivalence
  • Imaginaries and finitely presented functors
  • Elementary duality
  • Hulls of types and irreducible types
  • Interpretation functors
  • Stability
  • Ranks
  • Bibliography
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