Y.I. Manin mentions in a [recent interview](http://golem.ph.utexas.edu/category/2009/11/interview_with_manin.html#c028992 "link to n-category discussion")
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 standard 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](https://arxiv.org/abs/math/0502183 "arxiv") of Galois theory into the context of "brave new rings" and his [conference](http://folk.uio.no/rognes/yff/alexandra.html "website") last year, I wonder if themes discussed in [Kato's article](http://www.springerlink.com/index/p644163275773165.pdf "article") (e.g. reciprocity laws) have "brave new variants".


Edit: I found Greenlees' introductions ([1](https://arxiv.org/abs/math/0609452 "arxiv"), [2](https://arxiv.org/abs/math/0609453 "arxiv")) and Vogt's ["Introduction to Algebra over Brave New Rings"](http://hdl.handle.net/10338.dmlcz/701626) for getting an idea of the topological background very helpful.



What is the "intuition" behind "brave new algebra"?
I hardly know how to begin to reduce this subject to some kind of intuitive ideas, but here are some thoughts:


\* Adding homotopy to algebra allows for generalizations of familiar algebraic notions. For instance, a *topological commutative ring* is a commutative ring object in the category of spaces; it has addition and multiplication maps which satisfy the usual axioms such as associativity and commutativity. But instead, one might instead merely require that associativity and commutativity hold "up to all possible homotopies" (and we'll think of the homotopies as part of the structure). (It is hard to give the flavor of this if you haven't seen a definition of this sort.) This gives one possible definition of a "brave new commutative ring". 


\* What is really being generalized is not algebraic objects, but *derived categories* of algebraic objects. So if you have a brave new ring R, you don't really want to study the category of R-modules; rather, the proper object of study is the derived category of R-modules. If your ring R is an ordinary (cowardly old?) ring, then the derived category of R-modules is equivalent to the classical derived category of R.


If you want to generalize some classical algebraic notion to the new setting, you usually first have to figure out how to describe it in terms of derived notions; this can be pretty non-trivial in some cases, if not impossible. (For instance, I don't think there's any good notion of a subring of a brave new ring.)


\* As for Manin's remarks: the codification of these things has being an ongoing process for at least 40 years. It seems we've only now reached the point where these ideas are escaping homotopy theory and into the broad stream of mathematics. It will probably take a little while longer before things are so well codified that brave new rings get introduced in the grade school algebra curriculum, so the process certainly isn't over yet!


