This course M721 was given in Fall 2019 at Indiana University by Jim Davis. It investigates the algebraic topology of diagrams of spaces: functors from a fixed category to the category of topological spaces, generalizing the notion of a G-space. It is a panoramic view of categories, algebraic topology, simplicial methods, spectra, and algebraic K-theory. Exercises for the course are found here. The basic reference for the course is the paper Davis-Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory. Question and comments about the course are welcome.
It will be accessible to anyone who his familiar with algebraic topology at the level of Davis-Kirk, Lecture Notes in Algebraic Topology and, in particular, to those familiar with fibrations and cofibrations. Lück has numerous surveys on the subject of isomorphism conjectures on his webpage. Other useful references are May, A Concise Course in Algebraic Topology and Riehl, Category Theory in Context.