I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the subject and similar questions I have seen. It also seems that some of those qualified to answer are much more active here than on math.SE.
Question: What are some references that would supply me with the algebra, algebraic topology, category theory, etc. I need to get into the literature on homotopical group theory? If that is too broad then at least as a starting point, what would let me read Part III: Fusion and homotopy theory from Fusion systems in algebra and topology by Aschbacher, Kessar, and Oliver?
I have a general sense of what I need and I am aware of standard references for some of the background material, but there are many components to this and I would appreciate sources that may be especially relevant to the direction I want to go in (e.g. an algebraic topology text that has a good treatment of classifying spaces).
My background: I am not pursuing a career in mathematics but I have been teaching myself math for a number of years now. I have always connected most with group theory and topology and have recently become deeply fascinated by homotopical group theory (i.e. the study of a group via the homotopy theory of its classifying space). I would like to begin focusing my studies in that direction and could use some help with where to direct my efforts.
I have taken introductory courses in real analysis and algebra and have taught myself some linear algebra, graph theory, point-set topology, ring theory, module theory, and field theory, at least to the point of being able to work with the basics. Additionally, I have gone a bit further into group theory (e.g. group actions, Sylow theory, solvable and nilpotent groups, a tiny bit of fusion) and have done a little bit of algebraic topology (basics of homotopy and singular homology) from Rotman's text. I do have access to a university library, at least once the campus reopens.
Some specific concepts I find especially interesting relate to cellularization of classifying spaces (e.g. Flores and Scherer 2007, Cellularization of classifying spaces and fusion properties of finite groups) and homotopy group extensions (e.g. Broto and Levi 2002, On spaces of self-homotopy equivalences of p-completed classifying spaces of finite groups and homotopy group extensions).
