Memoirs of the American Mathematical Society 2011; 109 pp; softcover Volume: 210 ISBN10: 0821847678 ISBN13: 9780821847671 List Price: US$74 Individual Members: US$44.40 Institutional Members: US$59.20 Order Code: MEMO/210/987
 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 categorythe 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  Introduction
 Preadditive and additive categories
 Preadditive categories and their indcompletions
 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
 ppElimination of quantifiers
 Ultraproducts
 Pureinjectives and elementary equivalence
 Imaginaries and finitely presented functors
 Elementary duality
 Hulls of types and irreducible types
 Interpretation functors
 Stability
 Ranks
 Bibliography
