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 Gspace. It is a panoramic view of categories, algebraic topology, simplicial methods, spectra, and algebraic Ktheory. Exercises for the course are found here. The basic reference for the course is the paper DavisLÃ¼ck, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K and LTheory. Question and comments about the course are welcome.
It will be accessible to anyone who his familiar with algebraic topology at the level of DavisKirk, Lecture Notes in Algebraic Topology and, in particular, to those familiar with fibrations and cofibrations. LÃ¼ck 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.

From James Davis
The nonconnective Kspectrum of a small additive category constructed by PedersenWeibel, A Nonconnective Delooping of Algebraic Ktheory. 


From James Davis
The first three weeks of the course were not filmed (handwritten notes available upon request). Topics included the definition of a GCWcomplex (built from cells of… 
From James Davis
One can find a lot of info on Compactly Generated Weak Hausdorff spaces on the web (especially the notes of Strickland), but one can start with the books of May and… 
From James Davis
Equivariant homology theories were defined in Lück, Chern characters for proper equivariant homology theories and applications to K and Ltheory, but… 

From James Davis
For homotopy colimits and limits, I like the approach of DavisLück the best. The classical reference is BousfieldKan, Homotopy Limits, Completions and… 
From James Davis
The proof of the ladder and gluing lemmas can be found in MayPonto More Concise Algebraic Topology. 
From James Davis
References for this material are given in the book by Strom mentioned above as well as MunsonVolic Cubical Homotopy Theory. I don't know the geodesic route to… 
From James Davis
A good reference for exactness properties of classical colimits and limits is Weibel's book An Introduction to Homological Algebra. 
