By Harold Simmons

Type concept presents a normal conceptual framework that has proved fruitful in matters as various as geometry, topology, theoretical desktop technology and foundational arithmetic. here's a pleasant, easy-to-read textbook that explains the basics at a degree appropriate for beginners to the topic. starting postgraduate mathematicians will locate this e-book an outstanding creation to the entire fundamentals of type thought. It provides the elemental definitions; is going in the course of the numerous linked gadgetry, resembling functors, normal differences, limits and colimits; after which explains adjunctions. the cloth is slowly built utilizing many examples and illustrations to light up the innovations defined. Over two hundred routines, with options on hand on-line, support the reader to entry the topic and make the publication excellent for self-study. it may possibly even be used as a prompt textual content for a taught introductory direction.

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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 standard 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 type and Comma different types over a Category
CHAPTER 3. exclusive MORPHISMS AND OBJECTS
three. 1 exclusive Morphisms
three. 2 amazing 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 renovation of express Properties
four. three The Feeble Functor and opposite Quotient Functor
CHAPTER 5. normal alterations AND EQUIVALENCES
five. 1 usual adjustments and Their Compositions
five. 2 Equivalence of different types and Skeletons
five. three Functor Categories
five. four normal differences 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
7. eight vulnerable Adjoints
APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS
APPENDIX . ALGEBRAIC FUNCTORS
APPENDIX 3. TOPOLOGICAL FUNCTORS
Bibliography
Index

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Additional info for An Introduction to Category Theory

Example text

Don’t confuse this with the monoid R. For each r ∈ R there is an arrow r ✲ and again this has no internal structure. In other words the arrows of the category are the elements of R. Composition of arrows is just the carried operation of R. r ✲ s ✲ ✲ s ◦ r = sr The identity arrow id =1 is just the unit of R. This construction does produce a category because the operation on R is associative and 1 is a unit. On its own this example is rather trite, but later we will add to it to illustrate several aspects of category theory.

We meet notions such as diagram monic epic split monic split epic isomorphism initial final wedge product coproduct equalizer coequalizer pullback pushout universal solution some of which are discussed only informally. All of these notions are important, and have to be put somewhere in the book. It is more convenient to have them together in one place, and here seems the ‘logical’ place to put them. However, that does not mean you should plod through this chapter section by section. I suggest you get a rough idea of the notions involved, and then go to Chapter 3 (which discusses more important ideas).

4 Consider a composible pair of arrows. A m ✲ B n ✲ C Show that if both m and n are monic, then so is the composite n ◦ m. Show that if the composite n ◦ m is monic, then so is m. Find an example where the composite n ◦ m is monic but n is not. State the corresponding results for epics. Obtain similar results (where possible) for the other classes of arrows discussed in this section. 5 Consider the category Mon of monoids, and view N and Z as additively written monoids. Show that the insertion N ⊂ e ✲ Z is epic.

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