# Accessible Categories: The Foundations of Categorical Model Theory

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*Robert Paré; Michael Makkai*

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

## Accessible Categories: The Foundations of Categorical Model Theory

- Contents vii8 free
- Introduction 110 free
- Chapter 1: Preliminaries 1120
- Chapter 2: Accessible categories and functors 1726
- Chapter 3: Sketches and logic 3948
- Chapter 4: Sketching accessible categories 6776
- Chapter 5: Limits and Colimits of accessible categories 97106
- Chapter 6: Limits and Colimits in accessible categories 141150
- References 165174
- Index 167176
- Glossary of notation 173182