CATEGORIES AND SHEAVES

By W. BESAY

CONTENTS INTRODUCTION THE LANGUAGE OF CATEGORIES PRELIMINARIES: SETS AND UNIVERSES CATEGORIES AND FUNCTORS MORPHISMS OF FUNCTORS THE YONEDA LEMMA ADJOINT FUNCTORS EXERCISES LIMITS EXAMPLES KAN EXTENSION OF FUNCTORS INDUCTIVE LIMITS IN THE CATEGORY SET CO?NAL FUNCTORS IND-LIM AND PRO-LIM YONEDA EXTENSION OF FUNCTORS FILTRANT LIMITS FILTRANT INDUCTIVE LIMITS IN THE CATEGORY SET FILTRANT CATEGORIES EXACT FUNCTORS CATEGORIES ASSOCIATED WITH TWO FUNCTORS TENSOR CATEGORIES PROJECTORS RINGS, MODULES AND MONADS GENERATORS AND REPRESENTABILITY STRICT MORPHISMS STRICTLY GENERATING SUBCATEGORIES INDIZATION OF CATEGORIES INDIZATION OF CATEGORIES AND FUNCTORS REPRESENTABLE IND-LIMITS INDIZATION OF CATEGORIES ADMITTING INDUCTIVE LIMITS FINITE DIAGRAMS IN IND(C) LOCALIZATION LOCALIZATION OF CATEGORIES LOCALIZATION OF SUBCATEGORIES LOCALIZATION OF FUNCTORS INDIZATION AND LOCALIZATION ADDITIVE AND ABELIAN CATEGORIES GROUP OBJECTS ADDITIVE CATEGORIES ABELIAN CATEGORIES INJECTIVE OBJECTS RING ACTION INDIZATION OF ABELIAN CATEGORIES EXTENSION OF EXACT FUNCTORS ?-ACCESSIBLE OBJECTS AND F-INJECTIVE OBJECTS CARDINALS ?-?LTRANT CATEGORIES AND ?-ACCESSIBLE OBJECTS ?-ACCESSIBLE OBJECTS AND GENERATORS QUASI-TERMINAL OBJECTS F-INJECTIVE OBJECTS APPLICATIONS TO ABELIAN CATEGORIES TRIANGULATED CATEGORIES 10 1 TRIANGULATED CATEGORIES 10 2 LOCALIZATION OF TRIANGULATED CATEGORIES 10 3 LOCALIZATION OF TRIANGULATED FUNCTORS 10 4 EXTENSION OF COHOMOLOGICAL FUNCTORS 10 5 THE BROWN REPRESENTABILITY THEOREM COMPLEXES IN ADDITIVE CATEGORIES 11 1 DI?ERENTIAL OBJECTS AND MAPPING CONES 11 2 THE HOMOTOPY CATEGORY 11 3 COMPLEXES IN ADDITIVE CATEGORIES 11 4 SIMPLICIAL CONSTRUCTIONS 11 5 DOUBLE COMPLEXES 11 6 BIFUNCTORS 11 7 THE COMPLEX HOM COMPLEXES IN ABELIAN CATEGORIES 12 1 THE SNAKE LEMMA 12 2 ABELIAN CATEGORIES WITH TRANSLATION 12 3 COMPLEXES IN ABELIAN CATEGORIES 12 4 EXAMPLE: KOSZUL COMPLEXES 12 5 DOUBLE COMPLEXES DERIVED CATEGORIES 13 1 DERIVED CATEGORIES 13 2 RESOLUTIONS 13 3 DERIVED FUNCTORS 13 4 BIFUNCTORS UNBOUNDED DERIVED CATEGORIES 14 1 DERIVED CATEGORIES OF ABELIAN CATEGORIES WITH TRANSLATION 14 2 THE BROWN REPRESENTABILITY THEOREM 14 3 UNBOUNDED DERIVED CATEGORY 14 4 LEFT DERIVED FUNCTORS INDIZATION AND DERIVATION OF ABELIAN CATEGORIES 15 1 INJECTIVE OBJECTS IN IND(C) 15 2 QUASI-INJECTIVE OBJECTS 15 3 DERIVATION OF IND-CATEGORIES 15 4 INDIZATION AND DERIVATION GROTHENDIECK TOPOLOGIES 16 1 SIEVES AND LOCAL EPIMORPHISMS 16 2 LOCAL ISOMORPHISMS 16 3 LOCALIZATION BY LOCAL ISOMORPHISMS SHEAVES ON GROTHENDIECK TOPOLOGIES 17 1 PRESITES AND PRESHEAVES 17 2 SITES 17 3 SHEAVES 17 4 SHEAF ASSOCIATED WITH A PRESHEAF 17 5 DIRECT AND INVERSE IMAGES 17 6 RESTRICTION AND EXTENSION OF SHEAVES 17 7 INTERNAL HOM ABELIAN SHEAVES 18 1 R-MODULES 18 2 TENSOR PRODUCT AND INTERNAL HOM 18 3 DIRECT AND INVERSE IMAGES 18 4 DERIVED FUNCTORS FOR HOM AND HOM 18 5 FLATNESS 18 6 RINGED SITES 18 7 ?CECH COVERINGS STACKS AND TWISTED SHEAVES 19 1 PRESTACKS 19 2 SIMPLY CONNECTED CATEGORIES 19 3 SIMPLICIAL CONSTRUCTIONS 19 4 STACKS 19 5 MORITA EQUIVALENCE 19 6 TWISTED SHEAVES REFERENCES INTRODUCTION THE AIM OF THIS BOOK IS TO DESCRIBE THE TOPICS OUTLINED IN THE PREFACE, CAT- EGORIES, HOMOLOGICAL ALGEBRA AND SHEAVES WE ALSO PRESENT THE MAIN FEATURES AND KEY RESULTS IN RELATED TOPICS WHICH AWAIT A SIMILAR FULL-SCALE TREATMENT SUCH AS, FOR EXAMPLE, TENSOR CATEGORIES, TRIANGULATED CATEGORIES, STACKS THE GENERAL THEORY OF CATEGORIES AND FUNCTORS, WITH EMPHASIS ON INDUCTIVE AND PROJECTIVE LIMITS, TENSOR CATEGORIES, REPRESENTABLE FUNCTORS, IND-OBJECTS AND LOCALIZATION IS DEALT WITH IN CHAPS 1–7 HOMOLOGICAL ALGEBRA, INCLUDING ADDITIVE, ABELIAN, TRIANGULATED AND DERIVED CATEGORIES, IS TREATED IN CHAPS 8–15 CHAPTER 9 PROVIDES THE TOOLS (USING TRANS- ?NITE INDUCTION) WHICH WILL BE USED LATER FOR PRESENTING UNBOUNDED DERIVED CATEGORIES SHEAF THEORY IS TREATED IN CHAPS 16–19 IN THE GENERAL FRAMEWORK OF GRO- THENDIECK TOPOLOGIES IN PARTICULAR, THE RESULTS OF CHAP 14 ARE APPLIED TO THE STUDY OF THE DERIVED CATEGORY OF THE CATEGORY OF SHEAVES ON A RINGED SITE WE ALSO SKETCH AN APPROACH TO THE MORE SOPHISTICATED SUBJECT OF STACKS (ROUGHLY SPEAKING, SHEAVES WITH VALUES IN THE 2-CATEGORY OF CATEGORIES) AND INTRODUCE THE IMPORTANT NOTION OF TWISTED SHEAVES OF NECESSITY WE HAVE EXCLUDED MANY EXCITING DEVELOPMENTS AND APPLICATIONS SUCH AS N-CATEGORIES, OPERADS, A?-CATEGORIES, MODEL CATEGORIES, AMONG OTHERS WITHOUT DOUBT THESE NEW AREAS WILL SOON BE INTENSIVELY TREATED IN THE LITER- ATURE, AND IT IS OUR HOPE THAT THE PRESENT WORK WILL PROVIDE A BASIS FOR THEIR UNDERSTANDING WE NOW PROCEED TO A MORE DETAILED OUTLINE OF THE CONTENTS OF THE BOOK CHAPTER 1 WE BEGIN BY DE?NING THE BASIC NOTIONS OF CATEGORIES AND FUNC- TORS, ILLUSTRATED WITH MANY CLASSICAL EXAMPLES THERE ARE SOME SET–THEORETICAL DANGERS AND TO AVOID CONTRADICTIONS, WE WORK IN A GIVEN UNIVERSE UNIVERSES ARE PRESENTED AXIOMATICALLY, REFERRING TO [64] FOR A MORE DETAILED TREATMENT AMONG OTHER CONCEPTS INTRODUCED IN THIS CHAPTER ARE MORPHISMS OF FUNCTORS, EQUIVALENCES OF CATEGORIES, REPRESENTABLE FUNCTORS, ADJOINT FUNCTORS AND SO ON WE INTRODUCE IN PARTICULAR THE CATEGORY FCT(I, C) OF FUNCTORS FROM A SMALL