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This book is a textbook in basic category theory, written specifically to be read by researchers and students in computing science. The authors expound the constructions basic to category theory in the context of examples and applications to computing science. Some categorical ideas and constructions are already used heavily in computing sciences and many of these use are described. Other ideas, in particular the concept of adjoint have not appeared as widely in the computing science literature. The authors give an elementary exposition of those ideas they believe to be basic categorical tools, with pointers to possible application.

This new edition contains all the material from the first and second editions, including the four chapters excised from the second edition and the solutions to all the exercises, as well as added material on factorization systems, monoidal categories, and other topics. All errors known to the authors have been corrected.

Michael Barr is Peter Redpath Professor in the Department of Mathematics and Statistics at McGill University in Montreal, Quebec. Charles Wells is Professor of Mathematics at Case Western Reserve University in Cleveland, Ohio.

Preface
1 Preliminaries
1.1 Sets
1.2 Functions
1.3 Graphs
1.4 Homomorphisms of graphs
2 Categories
2.1 Basic definitions
2.2 Functional programming languages as categories
2.3 Mathematical structures as categories
2.4 Categories of sets with structure
2.5 Categories of algebraic structures
2.6 Constructions on categories
2.7 Properties of objects and arrows in a category
2.8 Monomorphisms and subobjects
2.9 Other types of arrow
2.10 Factorization systems
3 Functors
3.1 Functors
3.2 Actions
3.3 Types of functors
3.4 Equivalences
3.5 Quotient categories
4 Diagrams, naturality and sketches
4.1 Diagrams
4.2 Natural transformations
4.3 Natural transformations between functors
4.4 The Godement calculus of natural transformations
4.5 The Yoneda Lemma and universal elements
4.6 Linear sketches (graphs with diagrams)
4.7 Linear sketches with constants: initial term models
4.8 2-categories
5 Products and sums
5.1 The product of two objects in a category
5.2 Notation for and properties of products
5.3 Finite products
5.4 Sums
5.5 Natural numbers objects
5.6 Deduction systems as categories
5.7 Distributive categories
6 Cartesian closed categories
6.1 Cartesian closed categories
6.2 Properties of cartesian closed categories
6.3 Typed λ-calculus
6.4 λ-calculus to category and back
6.5 Arrows vs. terms
6.6 Fixed points in cartesian closed categories
7 Finite product sketches
7.1 Finite product sketches
7.2 The sketch for semigroups
7.3 Notation for FP sketches
7.4 Arrows between models of FP sketches
7.5 The theory of an FP sketch
7.6 Initial term models for FP sketches
7.7 Signatures and FP sketches
8 Finite discrete sketches
8.1 Sketches with sums
8.2 The sketch for fields
8.3 Term algebras for FD sketches
9 Limits and colimits
9.1 Equalizers
9.2 The general concept of limit
9.3 Pullbacks
9.4 Coequalizers
9.5 Cocones
9.6 More about sums
9.7 Unification as coequalizer
9.8 Properties of factorization systems
10 More about sketches
10.1 Finite limit sketches
10.2 Initial term models of FL sketches
10.3 The theory of an FL sketch
10.4 General definition of sketch
11 The category of sketches
11.1 Homomorphisms of sketches
11.2 Parametrized data types as pushouts
11.3 The model category functor
12 Fibrations
12.1 Fibrations
12.2 The Grothendieck construction
12.3 An equivalence of categories
12.4 Wreath products
13 Adjoints
13.1 Free monoids
13.2 Adjoints
13.3 Further topics on adjoints
13.4 Locally cartesian closed categories
14 Algebras for endofunctors
14.1 Fixed points for a functor
14.2 Recursive categories
14.3 Triples
14.4 Factorizations of a triple
14.5 Scott domains
15 Toposes
15.1 Definition of topos
15.2 Properties of toposes
15.3 Is a two-element poset complete?
15.4 Presheaves
15.5 Sheaves
15.6 Fuzzy sets
15.7 External functors
15.8 The realizability topos
16 Categories with monoidal structure
16.1 Closed monoidal categories
16.2 Properties of A −◦ C
16.3 ∗-autonomous categories
16.4 The Chu construction
Solutions to the exercises
Solutions for Chapter 1
Solutions for Chapter 2
Solutions for Chapter 3
Solutions for Chapter 4
Solutions for Chapter 5
Solutions for Chapter 6
Solutions for Chapter 7
Solutions for Chapter 8
Solutions for Chapter 9
Solutions for Chapter 10
Solutions for Chapter 11
Solutions for Chapter 12
Solutions for Chapter 13
Solutions for Chapter 14
Solutions for Chapter 15
Solutions for Chapter 16
Bibliography
Index
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