I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of category theory and don't really know much about what the applications of the topics and results in this chapter actually are. Since this reading was assigned for a seminar class where we give presentations on "seminal papers in algebraic topology", I'm assuming there is quite a bit going on here that I'm not aware of. I'm looking for some advice, comments, etc. that could help me get a better idea of the relevance of these topics and what role they played in the development of K-theories and other areas of algebraic topology over the last 40 years or so. Anything will help but I'm specifically looking for concrete examples and applications as well as anything that can help me frame this work historically. Some of the main topics in the chapter are
- Characterizing categories of modules
- "faithfully projective objects"
- R-categories: Right continuous functors
- (pre-)equivalence data
- Constructing an equivalence from a module
- Autoequivalences
- The Picard group
Much thanks ahead of time for any comments. Literally anything you might have to say that seems remotely relevant would be appreciated at this point (I've only had yesterday and until 4pm est today to prepare...)