This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.
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"A well-organized introduction to categorical methods of lambda calculus...The individual topics are treated in a uniform, balanced style from introduction, syntax, to categorical models and categorical-type theory correspondence." Jiri Adamek, Mathematical Reviews
Book Description
This explanation of the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories introduces the reader to ordered set theory, lattices and domains, providing many examples for an introduction to category theory.
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9 of 10 people found the following review helpful:
5.0 out of 5 stars
Excellent introduction to categories for computer scientists,
By peter@opus.geg.mot.com (Fountain Hills, Arizona) - See all my reviews
This review is from: Categories for Types (Studies in Emotion and Social Interaction) (Hardcover)
The book gives you all of the cateogry theory you need to study type theory. The examples are from domains that are comfortable for computer scientists. The difficult proofs are given in great detail, while other books often gloss over the details.
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Inside This Book
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First Sentence:
DISCUSSION 1 .1.1 We shall begin by giving an informal description of some of the topics which appear in Chapter 1. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
functional type theory, binary product types, algebraic complete lattice, raw operator, category with finite products, epic morphism, reindexing functors, mediating morphism, split monic, categorical semantics, presentable category, directed colimit, classifying category, raw terms, algebraic signature, preserves finite products, categories with finite products, filtered colimits, order polymorphism, proved terms, raw types, inductive subsets, indexed categories, adjoint functor theorem, directed poset Key Phrases - Capitalized Phrases (CAPs): (learn more)
Polymorphism Equations, Further Exercises, Polymorphism Terms, Function Terms, Binary Product Terms, Polymorphism Types, Term Theorems Generated New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
DISCUSSION 1 .1.1 We shall begin by giving an informal description of some of the topics which appear in Chapter 1. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
functional type theory, binary product types, algebraic complete lattice, raw operator, category with finite products, epic morphism, reindexing functors, mediating morphism, split monic, categorical semantics, presentable category, directed colimit, classifying category, raw terms, algebraic signature, preserves finite products, categories with finite products, filtered colimits, order polymorphism, proved terms, raw types, inductive subsets, indexed categories, adjoint functor theorem, directed poset Key Phrases - Capitalized Phrases (CAPs): (learn more)
Polymorphism Equations, Further Exercises, Polymorphism Terms, Function Terms, Binary Product Terms, Polymorphism Types, Term Theorems Generated New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 12
books:
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22
books
cite this book:
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- Stone Spaces (Cambridge Studies in Advanced Mathematics) by Peter T. Johnstone in Back Matter
- Basic Category Theory for Computer Scientists (Foundations of Computing) by Benjamin C. Pierce in Back Matter
- General Lattice Theory (Second Edition) by George Grätzer in Back Matter
- Proofs and Types (Cambridge Tracts in Theoretical Computer Science) by Jean-Yves Girard in Back Matter
- Computer Science by David Woodhouse in Back Matter
See all 12 books this book cites
- Algebraic and Coalgebraic Methods in the Mathematics of Program Construction: International Summer School and Workshop, Oxford, UK, April 10-14, 2000, ... Lectures (Lecture Notes in Computer Science) by Roland Backhouse on page 19, and page 195
- Computer Science Logic: 8th Workshop, CSL '94, Kazimierz, Poland, September 25 - 30, 1994. Selected Papers (Lecture Notes in Computer Science) by Leszek Pacholski on page 353
- Generic Programming: Advanced Lectures (Lecture Notes in Computer Science) by Roland Backhouse on page 176
- The Agile Virtual Enterprise: Cases, Metrics, Tools by Ted Goranson in Back Matter
- Logics for Concurrency: Structure versus Automata (Lecture Notes in Computer Science) by Faron Moller on page 39
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