By Viakalathur Sankrithi Krishnan

CONTENTS
========+

Preface
CHAPTER ONE. fundamentals FROM ALGEBRA AND TOPOLOGY
1.1 Set Theory
1.2 a few regular Algebraic Structures
1.3 Algebras in General
1.4 Topological Spaces
1.5 Semimetric and Semiuniform Spaces
1.6 Completeness and the Canonical Completion
CHAPTER . different types, DEFINITIONS, AND EXAMPLES
2.1 Concrete and basic Categories
2.2 Subcategories and Quotient Categories
2.3 items and Coproducts of Categories
2.4 the twin class and Duality of Properties
2.5 Arrow classification and Comma different types over a Category
CHAPTER 3. distinct MORPHISMS AND OBJECTS
3.1 distinctive Morphisms
3.2 distinct Objects
3.3 Equalizers and Coequalizers
3.4 consistent Morphisms and Pointed Categories
3.5 Separators and Coseparators
CHAPTER 4. varieties of FUNCTORS
4.1 complete, devoted, Dense, Embedding Functors
4.2 mirrored image and maintenance of express Properties
4.3 The Feeble Functor and opposite Quotient Functor
CHAPTER 5. average alterations AND EQUIVALENCES
5.1 common changes and Their Compositions
5.2 Equivalence of different types and Skeletons
5.3 Functor Categories
5.4 typical alterations 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
6.3 Factorizations of Morphisms
6.4 Completeness
CHAPTER SEVEN. ADJOINT FUNCTORS
7.1 the trail Category
7.2 Adjointness
7.3 Near-equivalence and Adjointness
7.4 Composing and Resolving Shortest Paths or Adjoints
7.5 Adjoint Functor Theorems
7.6 Examples of Adjoints
7.7 Monads
7.8 susceptible Adjoints
APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS
APPENDIX . ALGEBRAIC FUNCTORS
APPENDIX 3. TOPOLOGICAL FUNCTORS
Bibliography
Index

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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 usual Algebraic Structures
1. three Algebras in General
1. four Topological Spaces
1. five Semimetric and Semiuniform Spaces
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2. 1 Concrete and normal Categories
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four. 1 complete, trustworthy, Dense, Embedding Functors
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CHAPTER 5. average modifications AND EQUIVALENCES
five. 1 ordinary differences and Their Compositions
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CHAPTER SIX. LIMITS, COLIMITS, COMPLETENESS, COCOMPLETENESS
6. 1 Predecessors and boundaries of a Functor
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Bibliography
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Extra resources for Introduction to Category Theory

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4. Check that (X, e) is a semigroup in which each element is both a left unit and a right zero if we define x e y = y for each pair of elements x, y from EXERCISES AND REMARKS ON CHAPTER I 23 X. Show that this semigroup can have no right unit (unless X has just one element). 5. Show that a semigroup (X, 9) with a left unit element e gives rise to a group (X, 9, ( )') if the unary operation on X is so defined that (x)' 9 x = e for each x of X; that is, if each x of X has a left-inverse (x)' relative to e.

EXERCISES AND REMARKS ON CHAPTER I 29 Case 2: h = n - l. This means that g(x, z 1) + · · · + g(zn_ 2 , Zn- 1) < s/2 and g(Zn-•• y) < s; as before, the induction hypothesis gives g(x, zn_ 1) < s also. Hence again we get 1/2 · g(x, y) s; s. Case 3: 0 < h < n - l. There are three parts in the sum, such that the first and third have a value

Using the various types of topological spaces (such as T0-, T3-, CR-spaces) as objects but taking for morphisms either the "open maps" or the "closed maps," we get whole new series of categories of spaces, which we denote by (T0 , 0), (CR, Cl), etc. (A map f of a space (X, G) in another (Y, H) is called an open (a closed) map if it takes open (closed) sets of (X, G) to open (closed) sets in (Y, H)). Show that (T2 'COM) is a subcategory of (T2 ', Cl). Is it a full subcategory? 2. For the semiuniform, quasiuniform, or uniform spaces as objects we may take the morphisms to be maps between two such spaces that are merely continuous under the induced topologies, not uniformly continuous.

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