This meeting will cover 2.3.C (page 79) through 2.4.3 (page 83). The most important point will be to see how the category of presheaves of abelian groups on an topological space X forms an abelian category.
- 2.3.C. How is the “sheaf Hom” defined? What are the restriction maps? Why is it a sheaf?
- 2.3.C-Warning: Can you find an example showing sheaf-hom does not commute with taking stalks?
- We will go over in detail the arguments showing that the category of presheaves of abelian groups on a space X forms an abelian category. Start with the defining properties of an additive category. How are the sets of morphisms viewed as abelian groups? What is the 0-object? What is a product of two presheaves?
- What is the kernel and cokernel in this category? How are restriction maps defined on these sheaves?
- Complete the verification that this category is abelian.
- 2.3.G. Specialize the definition of exactness to this abelian category.
- 2.3.H. Characterize exactness in terms of sections over each U.
- Exercise 2.3.I.
- Exercise 2.3.J.
- Exercise 2.4.A.
- Exercise 2.4.B.