By D. E. Rydeheard

CONTENTS

========

1 Introduction

1.1 The contents

1.2 Accompanying texts

1.2.1 Textbooks on classification theory

1.2.2 ML references and availability

1.2.3 a variety of textbooks on practical programming

1.3 Acknowledgements

2 practical Programming in ML

2.1 Expressions, values and environments

2.2 Functions

2.2.1 Recursive definitions

2.2.2 better order functions

2.3 Types

2.3.1 Primitive types

2.3.2 Compound types

2.3.3 kind abbreviation

2.4 style polymorphism

2.5 Patterns

2.6 Defining types

2.7 summary types

2.8 Exceptions

2.9 different facilities

2.10 Exercises

3 different types and Functors

3.1 Categories

3.1.1 Diagram chasing

3.1.2 Subcategories, isomorphisms, monics and epis

3.2 Examples

3.2.1 units and finite sets

3.2.2 Graphs

3.2.3 Finite categories

3.2.4 kinfolk and partial orders

3.2.5 Partial orders as categories

3.2.6 Deductive systems

3.2.7 common algebra: phrases, algebras and equations

3.2.8 units with constitution and structure-preserving arrows

3.3 different types computationally

3.4 different types as values

3.4.1 the class of finite sets

3.4.2 phrases and time period substitutions: the class T_tau^Fin

3.4.3 A finite category

3.5 Functors

3.5.1 Functors computationally

3.5.2 Examples

3.6 Duality

3.7 An assessment*

3.8 Conclusion

3.9 Exercises

4 Limits and Colimits

4.1 Definition via universality

4.2 Finite colimits

4.2.1 preliminary objects

4.2.2 Binary coproducts

4.2.3 Coequalizers and pushouts

4.3 Computing colimits

4.4 Graphs, diagrams and colimits

4.5 A basic development of colimits

4.6 Colimits within the class of finite sets

4.7 A calculation of pushouts

4.8 Duality and limits

4.9 Limits within the classification of finite sets

4.10 An program: operations on relations

4.11 Exercises

5 developing Categories

5.1 Comma categories

5.1.1 Representing comma categories

5.2 Colimits in comma categories

5.3 Calculating colimits of graphs

5.4 Functor categories

5.4.1 average transformations

5.4.2 Functor categories

5.5 Colimits in functor categories

5.6 Duality and limits

5.7 summary colimits and limits*

5.7.1 summary diagrams and colimits

5.7.2 classification constructions

5.7.3 listed colimit structures

5.7.4 Discussion

5.8 Exercises

6 Adjunctions

6.1 Definitions of adjunctions

6.2 Representing adjunctions

6.3 Examples

6.3.1 flooring and ceiling capabilities: changing genuine numbers to integers

6.3.2 elements of a graph

6.3.3 loose algebras

6.3.4 Graph theory

6.3.5 Limits and colimits

6.3.6 Adjunctions and comma categories

6.3.7 Examples from algebra and topology

6.4 Computing with adjunctions

6.5 unfastened algebras

6.5.1 developing loose algebras

6.5.2 A program

6.5.3 An instance: transitive closure

6.5.4 different buildings of loose algebras

6.6 Exercises

7 Toposes

7.1 Cartesian closed categories

7.1.1 An instance: the class of finite sets

7.2 Toposes

7.2.1 An instance: the topos of finite sets

7.2.2 Computing in a topos

7.2.3 common sense in a topos

7.2.4 An instance: a three-valued logic

7.3 Conclusion

7.4 Exercises

8 A specific Unification Algorithm

8.1 The unification of terms

8.2 Unification as a coequalizer

8.3 On developing coequalizers

8.4 A specific program

9 developing Theories

9.1 Preliminaries

9.2 developing theories

9.3 Theories and institutions

9.4 Colimits of theories

9.5 Environments

9.6 Semantic operations

9.7 imposing a express semantics

10 Formal structures for class Theory

10.1 Formal facets of class theory

10.2 classification conception in OBJ

10.3 class concept in a kind theory

10.4 express info types

A ML Keywords

B Index of ML Functions

C different ML Functions

D solutions to Programming routines 231

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**Introduction to Category Theory**

CONTENTS

========+

Preface

CHAPTER ONE. fundamentals FROM ALGEBRA AND TOPOLOGY

1. 1 Set Theory

1. 2 a few average Algebraic Structures

1. three Algebras in General

1. four Topological Spaces

1. five Semimetric and Semiuniform Spaces

1. 6 Completeness and the Canonical Completion

CHAPTER . different types, DEFINITIONS, AND EXAMPLES

2. 1 Concrete and common Categories

2. 2 Subcategories and Quotient Categories

2. three items and Coproducts of Categories

2. four the twin classification and Duality of Properties

2. five Arrow class and Comma different types over a Category

CHAPTER 3. distinctive MORPHISMS AND OBJECTS

three. 1 exclusive Morphisms

three. 2 distinct Objects

three. three Equalizers and Coequalizers

three. four consistent Morphisms and Pointed Categories

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CHAPTER 4. forms of FUNCTORS

four. 1 complete, trustworthy, Dense, Embedding Functors

four. 2 mirrored image and upkeep of express Properties

four. three The Feeble Functor and opposite Quotient Functor

CHAPTER 5. typical modifications AND EQUIVALENCES

five. 1 typical ameliorations and Their Compositions

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five. three Functor Categories

five. four normal alterations for Feeble Functors

CHAPTER SIX. LIMITS, COLIMITS, COMPLETENESS, COCOMPLETENESS

6. 1 Predecessors and bounds of a Functor

6. 2 Successors and Colimits of a Functor

6. three Factorizations of Morphisms

6. four Completeness

CHAPTER SEVEN. ADJOINT FUNCTORS

7. 1 the trail Category

7. 2 Adjointness

7. three Near-equivalence and Adjointness

7. four Composing and Resolving Shortest Paths or Adjoints

7. five Adjoint Functor Theorems

7. 6 Examples of Adjoints

7. 7 Monads

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APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS

APPENDIX . ALGEBRAIC FUNCTORS

APPENDIX 3. TOPOLOGICAL FUNCTORS

Bibliography

Index

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**Sample text**

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