CATEGORY THEORY

By B. BEZII

CONTENTS CATEGORIES INTRODUCTION FUNCTIONS OF SETS DE?NITION OF A CATEGORY EXAMPLES OF CATEGORIES ISOMORPHISMS CONSTRUCTIONS ON CATEGORIES FREE CATEGORIES FOUNDATIONS: LARGE, SMALL, AND LOCALLY SMALL EXERCISES ABSTRACT STRUCTURES EPIS AND MONOS INITIAL AND TERMINAL OBJECTS GENERALIZED ELEMENTS SECTIONS AND RETRACTIONS PRODUCTS EXAMPLES OF PRODUCTS CATEGORIES WITH PRODUCTS HOM-SETS DUALITY THE DUALITY PRINCIPLE COPRODUCTS EQUALIZERS COEQUALIZERS GROUPS AND CATEGORIES GROUPS IN A CATEGORY THE CATEGORY OF GROUPS GROUPS AS CATEGORIES FINITELY PRESENTED CATEGORIES LIMITS AND COLIMITS SUBOBJECTS PULLBACKS PROPERTIES OF PULLBACKS LIMITS PRESERVATION OF LIMITS COLIMITS EXPONENTIALS EXPONENTIAL IN A CATEGORY CARTESIAN CLOSED CATEGORIES HEYTING ALGEBRAS EQUATIONAL DE?NITION ?-CALCULUS FUNCTORS AND NATURALITY CATEGORY OF CATEGORIES REPRESENTABLE STRUCTURE STONE DUALITY NATURALITY EXAMPLES OF NATURAL TRANSFORMATIONS EXPONENTIALS OF CATEGORIES FUNCTOR CATEGORIES EQUIVALENCE OF CATEGORIES EXAMPLES OF EQUIVALENCE 7 10 EXERCISES CATEGORIES OF DIAGRAMS SET-VALUED FUNCTOR CATEGORIES THE YONEDA EMBEDDING THE YONEDA LEMMA APPLICATIONS OF THE YONEDA LEMMA LIMITS IN CATEGORIES OF DIAGRAMS COLIMITS IN CATEGORIES OF DIAGRAMS EXPONENTIALS IN CATEGORIES OF DIAGRAMS TOPOI ADJOINTS PRELIMINARY DE?NITION HOM-SET DE?NITION EXAMPLES OF ADJOINTS ORDER ADJOINTS QUANTI?ERS AS ADJOINTS RAPL LOCALLY CARTESIAN CLOSED CATEGORIES ADJOINT FUNCTOR THEOREM 10 MONADS AND ALGEBRAS 10 1 THE TRIANGLE IDENTITIES 10 2 MONADS AND ADJOINTS 10 3 ALGEBRAS FOR A MONAD 10 4 COMONADS AND COALGEBRAS 10 5 ALGEBRAS FOR ENDOFUNCTORS 10 6 EXERCISES