By Yde Venema

It is a doctoral dissertation of Yde Venema below the supervision of prof. Johan van Benthem.

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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 common 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 classification and Comma different types over a Category

CHAPTER 3. extraordinary MORPHISMS AND OBJECTS

three. 1 unique Morphisms

three. 2 unusual Objects

three. three Equalizers and Coequalizers

three. four consistent Morphisms and Pointed Categories

three. five Separators and Coseparators

CHAPTER 4. sorts of FUNCTORS

four. 1 complete, trustworthy, Dense, Embedding Functors

four. 2 mirrored image and renovation of specific Properties

four. three The Feeble Functor and opposite Quotient Functor

CHAPTER 5. ordinary modifications AND EQUIVALENCES

five. 1 typical differences and Their Compositions

five. 2 Equivalence of different types and Skeletons

five. three Functor Categories

five. four typical 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

7. eight vulnerable Adjoints

APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS

APPENDIX . ALGEBRAIC FUNCTORS

APPENDIX 3. TOPOLOGICAL FUNCTORS

Bibliography

Index

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**Sample text**

By assumption we have v / / ( 0 ,77), and as / is a potential zigzag morphism we get a situation as showed in the right picture. By # |= N V 7 ( f ,(rj,r])), $ has a vv £ D with Hvvv and Vvvf'(rj,rj). We define /'O hO = V and set /'(£>£) as the unique diagonal if-successor of any/all of the f'(rjy£). It is straightforward to verify that with this definition the part of / ' defined up till now satisfies both conditions (1) and (2). 2. TWO-DIMENSIONAL CYLINDRIC MODAL LOGIC. D / \ tin 1 1 \v v,/ 1 1 Ki j / H V i, f'oo fi10 y / For the definition of /'(£, rj) (rj < £), we use the same trick as above to ensure /'(£, rj) £ D : as /'(£, £) is in D and f r(rj, £) is not, they cannot be identical.

In chapter 5 we will give a detailed account of how temporal logics of intervals fit in the two-dimensional picture. Many of the systems described above may be perceived as following a general trend in modal logic, namely of bridging the gap between the old-fashioned intensional framework, which is simple, elegant and has nice computational properties, and classical first order logic which is expressive and perhaps still more familiar. 1. INTRODUCTION. of the classical modal formalism are: Blackburn [19] and Gargov and Goranko [39] add ‘nominals’ or ‘names’ to the language, these being atomic propositions holding at unique possible worlds.

Section the standard correspondence map as defined for the similarity type. 3 for details) is not the same as 46 CHAPTER 3. THE SQUARE UNIVERSE. worlds consisting of a Carthesian square2 and the interpretation map having a fixed and uniform definition for this kind of universes. 2. (*>»)) I {((«>»). (»,»)) 1 {((u ,v ),(w ,x ),(y ,z)) | u = w ,v s* II & w II II 53 53 53 I(* ) 1(0 ) I(O ’) 1(0') 1 (0 ) 1(0) m m i(e ) I( o) = A two-dimensional or square model is a model based on a two-dimensional frame.

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