Derived Categories in Topology

The first section contains various basic facts on sheaves, including the definition of (hyper) cohomology, some standard associated spectral sequences and several versions of the celebrated de Rham Theorem. After briefly discussing the derived tensor product in the second section, we give an ample introduction to the direct and inverse images of sheaves under continuous mappings in section 3. The adjunction triangle is singled out in the forth section, since this is one of the recurrent tools used in these notes. The last section is devoted to the first properties of the local systems. These are the building blocks for more complicated sheaves and, in the same time, the sheaves were the marriage between algebra and topology is easily seen.