Y.I. Manin mentions in a recent interview the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I thought that is codified in e.g. Lurie’s articles. But I read only his survey on elliptic cohomology and some standart articles on symmetric spectra. Taking the quoted remark as indicator for me having missed to notice something, I’d like to read what others think about that, esp. what the intuition on “brave new algebra” is.
Edit: In view of Rognes' transfer of Galois theory into the context of "brave new rings" and his conference last year, I wonder if themes discussed in Kato's article (e.g. reciprocity laws) have "brave new variants"?
Edit: I found Greenlees' introductions (1, 2) and Vogt's "Introduction to Algebra over Brave New Rings" for getting an idea of the topological background very helpfull.
