By D. E. Rydeheard

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Introduction to Category Theory

CONTENTS
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Preface
CHAPTER ONE. fundamentals FROM ALGEBRA AND TOPOLOGY
1. 1 Set Theory
1. 2 a few ordinary 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 normal Categories
2. 2 Subcategories and Quotient Categories
2. three items and Coproducts of Categories
2. four the twin class and Duality of Properties
2. five Arrow type and Comma different types over a Category
CHAPTER 3. distinctive MORPHISMS AND OBJECTS
three. 1 extraordinary Morphisms
three. 2 extraordinary Objects
three. three Equalizers and Coequalizers
three. four consistent Morphisms and Pointed Categories
three. five Separators and Coseparators
CHAPTER 4. kinds of FUNCTORS
four. 1 complete, trustworthy, Dense, Embedding Functors
four. 2 mirrored image and maintenance of specific Properties
four. three The Feeble Functor and opposite Quotient Functor
CHAPTER 5. typical alterations AND EQUIVALENCES
five. 1 typical adjustments and Their Compositions
five. 2 Equivalence of different types and Skeletons
five. three Functor Categories
five. four common variations for Feeble Functors
CHAPTER SIX. LIMITS, COLIMITS, COMPLETENESS, COCOMPLETENESS
6. 1 Predecessors and boundaries 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
7. eight susceptible Adjoints
APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS
APPENDIX . ALGEBRAIC FUNCTORS
APPENDIX 3. TOPOLOGICAL FUNCTORS
Bibliography
Index

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Additional info for Computational Category Theory (Prentice-Hall International Series in Computer Science)

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1,true,6] is not a list in this sense. Any list may be built from the empty list by successively putting items on the front of the list. Let nil denote the empty list and :: the operation which takes an item and a list and creates a new list consisting of the old list with the item on the front. g. 1::(2::(3::nil)) is the list [1,2,3]. Consider the function append, which concatenates two lists end to end: append([1,3,2],[3,4]) = [1,3,2,3,4] This clearly acts independently of the type of the items in the lists and so is polymorphic.

Exercise∗ 8. More list processing We have already seen many functions on lists. However, this is but a small sample of a rich vein of functions which illustrate the utility of recursion as a concise and executable definition mechanism. Here are a few more suggestions for list processing functions. Clearly there are many similar functions which you may wish to encode. 1. The function which deletes all occurrences of a value from a list is defined as follows: fun delete(x,nil) = | delete(x,a::s) if x=a then else nil = delete(x,s) a::delete(x,s) Define the function which deletes the n-th occurrence of a value.

In languages where this is not the case we represent finite sets using other types such as arrays or lists. In the previous chapter we defined sets as an abstract type represented by linear lists. Using this representation, we denote by ’a Set the type of finite sets whose elements have type ’a. Arrows in FinSet are typed functions – triples consisting of two sets and a function between them. As a data type this is: datatype ’a Set_Arrow = set_arrow of (’a Set)*(’a->’a)*(’a Set) We now define four functions describing how finite sets and set arrows form a category.

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