By Yehoshua Bar-Hillel (Eds.)

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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 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 normal 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. extraordinary MORPHISMS AND OBJECTS
three. 1 amazing Morphisms
three. 2 unusual Objects
three. three Equalizers and Coequalizers
three. four consistent Morphisms and Pointed Categories
three. five Separators and Coseparators
CHAPTER 4. forms of FUNCTORS
four. 1 complete, trustworthy, Dense, Embedding Functors
four. 2 mirrored image and protection of express Properties
four. three The Feeble Functor and opposite Quotient Functor
CHAPTER 5. ordinary changes AND EQUIVALENCES
five. 1 traditional ameliorations and Their Compositions
five. 2 Equivalence of different types and Skeletons
five. three Functor Categories
five. four usual adjustments 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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Extra info for Mathematical Logic and Foundations of Set Theory, Proceedings of an International Colloquium Held Under the Auspices of The Israel Academy of Sciences and Humanities

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This construction appeared in Rogers 1967 where it was shown (in effect) that (11) i f r i s parametrized, closed under recursive substitutions, &, v, Vcc and Prewellordering (r),then Prewellordering (C'T) The prewellordering property was the key to the development of the theory of semi-hyperanalytic sets in Moschovakis 1967 and the theory of semi-hyperprojective sets in Moschovakis 1969. In each of these cases we can lift much of the theory of II: to these classes - and we can do more because of the stronger closure properties that we can utilize.

It turns out that in the context of A D , 0 ( ~ 2is) utterly huge. In this section we give a few results that suggest this. 7 Results of this type were first obtained (to the best of our knowledge) by H. Friedman and ourselves, independently, in the winter and spring of 1968. g. nxl) are order types of prewellorderings of R. g. that each subset of nxI is definable from a real. Our chief result was that (with A D and DC) each uk is the order type of a projective prewellordering of R and that larger ordinals, like n,, are order types of hyperanalytic prewellorderings.

If K is regular and (A, ( A * B ) n A ) < % ( K , A * B ) , then it is immediate that (A, A nA ) ~ < ( K , A ) , (A,B nA)<<(K, B). Similarly, if 5 < K and {A,,},, 5 and (A,*{As},,

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