This course is the third semester algebraic topology course at UVA. The basic topics will be a focus on the higher homotopy groups and an evaluation of the question "how can we understand spaces 'up to homotopy'? What do our invariants really measure?"

There will be a recurring theme of how chain complexes and the derived category of modules over a ring can model homotopy theory. This provides a computational way to check conjectures and to make sure statements make sense in a simpler, algebraic environment. We'll build on this, describing the language of model categories as a way to better understand the interplay between spaces and chain complexes.