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Accessible Categories: The Foundations of Categorical Model Theory
Michael Makkai and Robert Paré
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Contemporary Mathematics
1990; 176 pp; softcover
Volume: 104
Reprint/Revision History:
reprinted 1991
ISBN-10: 0-8218-5111-X
ISBN-13: 978-0-8218-5111-1
List Price: US$42
Member Price: US$33.60
Order Code: CONM/104
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Intended for category theorists and logicians familiar with basic category theory, this book focuses on categorical model theory, which is concerned with the categories of models of infinitary first order theories, called accessible categories. The starting point is a characterization of accessible categories in terms of concepts familiar from Gabriel-Ulmer's theory of locally presentable categories. Most of the work centers on various constructions (such as weighted bilimits and lax colimits), which, when performed on accessible categories, yield new accessible categories. These constructions are necessarily 2-categorical in nature; the authors cover some aspects of 2-category theory, in addition to some basic model theory, and some set theory. One of the main tools used in this study is the theory of mixed sketches, which the authors specialize to give concrete results about model theory. Many examples illustrate the extent of applicability of these concepts. In particular, some applications to topos theory are given.

Perhaps the book's most significant contribution is the way it sets model theory in categorical terms, opening the door for further work along these lines. Requiring a basic background in category theory, this book will provide readers with an understanding of model theory in categorical terms, familiarity with 2-categorical methods, and a useful tool for studying toposes and other categories.

Table of Contents

  • Preliminaries
  • \(\kappa\)-filtered colimits
  • \(\kappa\)-flat functors
  • Accessible categories and functors
  • \(\kappa\)-accessible categories
  • Small categories and accessibility
  • Raising the index of accessibility
  • Accessible functors
  • An equivalent definition of accessibility
  • Sketches and logic
  • Sketches
  • Logic
  • The downward Löwenheim-Skolem theorem
  • Examples
  • Sketching accessible categories
  • Preliminaries on \(2\)-categories
  • The canonical sketch associated with a \(\kappa\)-accessible category
  • Small sketches for an accessible category
  • Diagrams of accessible categories and diagrams of sketches
  • Axiomatizing an accessible subcategory
  • Limits and colimits of accessible categories
  • Colimits of sketches and limits of accessible categories
  • Some results concerning Grothendieck toposes
  • Accessible fibrations
  • Lax colimits of accessible categories
  • The powerful image of an accessible functor
  • Limits and colimits in accessible categories
  • Completeness and cocompleteness in accessible categories
  • Models of a sketch in an accessible category
  • Detectability of colimits
  • Completing an accessible category
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