An Introduction to Categories and Sheaves

"This text may be considered as an elementary introduction to the book [Categories and Sheaves - Masaki Kashiwara, Pierre Schapira] and in fact a few (as few as possible) difficult proofs, such as the Brown representability theorem, are omitted here, with the reader referred to that book. On the other hand, we study in some details sheaves on topological spaces and particularly on locally compact spaces, including duality, topics which are not treated in the book mentioned above." Preface Contents Introduction 1. The language of categories 1.1 Sets and maps 1.2 Modules and linear maps 1.3 Categories and functors 1.4 The Yoneda Lemma 1.5 Representable functors, adjoint functors Exercises 2. Limits 2.1 Products and coproducts 2.2 Kernels and cokernels 2.3 Limits 2.4 Fiber products and coproducts 2.5 Properties of limits 2.6 Filtered colimits Exercises 3. Localization 3.1 Localization of categories 3.2 Localization of subcategories 3.3 Localization of functors Exercises 4. Additive categories 4.1 Additive categories 4.2 Complexes in additive categories 4.3 Double complexes 4.4 The homotopy category 4.5 Simplicial constructions Exercises 5. Abelian categories 5.1 Abelian categories 5.2 Exact functors 5.3 Injective and projective objects 5.4 Generators and Grothendieck categories 5.5 Complexes in abelian categories 5.6 Double complexes in abelian categories 5.7 Koszul complexes Exercises 6. Triangulated categories 6.1 Triangulated categories 6.2 Triangulated and cohomological functors 6.3 Applications to the homotopy category 6.4 Localization of triangulated categories Exercises 7. Derived categories 7.1 Derived categories 7.2 Resolutions 7.3 Derived functors 7.4 Bifunctors 7.5 The Brown representability theorem Exercises 8. Sheaves on sites 8.1 Abelian sheaves on topological spaces: a short introduction 8.2 Presites and presheaves 8.3 Operations on presheaves 8.4 Grothendieck topologies 8.5 Sheaves 8.6 Sheaf associated with a presheaf 8.7 Operations on sheaves 8.8 Locally constant sheaves and glueing of sheaves Exercises 9. Derived categories of abelian sheaves 9.1 Abelian sheaves 9.2 Flat sheaves and injective sheaves 9.3 The derived category of sheaves 9.4 Modules over sheaves of rings 9.5 Ringed sites Exercises 10. Sheaves on topological spaces 10.1 Restriction of sheaves 10.2 Sheaves associated with a locally closed subset 10.3 Čech complexes for open coverings 10.4 Čech complexes for closed coverings 10.5 Flabby sheaves 10.6 Sheaves on the interval [0,1] 10.7 Invariance by homotopy 10.8 Action of groups 10.9 Cohomology of some classical manifolds Exercises 11. Duality on locally compact spaces 11.1 Proper direct images 11.2 c-soft sheaves 11.3 Derived proper direct images 11.4 The functor f^! 11.5 Orientation and duality on l0-manifolds 11.6 Cohomology of real and complex manifolds Exercises 12. Constructible sheaves and the subanalytic topology 12.1 Subanalytic subsets 12.2 Subanalytic geometry 12.3 Constructible sheaves 12.4 Subanalytic topology Bibliography