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Now I'm going to try to talk a bit about algebra in spectra, or "derived algebra," or "brave new algebra," or "higher algebra," or whatever you want, mainly at the request of @joyo_fresh. This is basically a continuation of this thread:
So what we came away with was that there was this new category of things called "spectra" which admitted interesting functors from pointed spaces (this Σ^∞ functor) and from abelian groups (the Eilenberg Mac Lane functor H).
And what's more, there were other spectra that didn't come from abelian groups or spaces at all, but which still give us cohomology theories, like complex K-theory, cobordism theories, elliptic cohomology, and algebraic K-theory of rings.
Then what we want to do now is ask: how much "algebra" can I do in this setting? Because, in a genralization of the "smash product" of pointed spaces, it turns out that Spectra is a symmetric monoidal (∞-)category!
The first thing to note is that the functor H:Ab⭢Spectra is lax symmetric monoidal. In other words, it's not the case that H(A⊗B)≃H(A)⊗H(B), but it is true that H of a (commutative) monoid in Ab is a (commutative) monoid in Spectra.
As a result, every Eilenberg MacLane spectrum HR, for R a ring, is a monoid object in Spectra, and we'd call it a "ring spectrum." One terminological note: monoids in Ab are rings, so monoids in Spectra are ring spectra, but HR is not, for instance, a RING object in Ab.
But, again, lots of spectra which aren't Eilenberg MacLane spectra are ring spectra. In particular the topological K-theory spectra KU and KO are both commutative rings, as well MG, where MG is the cobordism theory associated to G oriented bordisms.
It also happens that the sphere spectrum 𝕊=Σ^∞(S^0)={S^0, S¹, S²,...} is a commutative ring spectrum. In fact, 𝕊 is the monoidal unit in Spectra, so sometimes people call Spectra 𝕊-modules, and ring spectra 𝕊-algebras.
Really there's an enormous amount to say about all of this, and about a thousand different directions one could take a thread on the topic, so I don't think I'll say a whole lot more. But just a few quick hits:
1. John Rognes has generalized classical Galois theory to the setting of ring spectra, and this has proven really useful both theoretically and computationally. In particular, one can use non-discrete groups for Galois groups, and do Galois descent and Galois cohomology!
2. Lurie and others have written a whole lot about doing algebraic geometry in this setting. In particular, there is a really useful notion of "spectral scheme," and many classical AG concepts have been developed in this framework.
3. The whole world of chromatic homotopy theory can be thought of as investigating the "prime ideals" of the sphere spectrum 𝕊, which turn out to be the usual prime ideals of ℤ, plus for each prime p∊ℤ, a new list of prime ideals K(n) called the Morava K-theories.
(see this older thread I wrote for more on chromatic homotopy theory: )
3.5: One caveat though: the general theory of "ideals" DOES NOT WORK in spectra. It doesn't make sense to talk about ideals or prime ideals. What we can talk about is localizing and thick subcategories (and boy howdy we sure do talk about em...).
4. Some people feel that working in spectra some kind of approach to working over the "field with one element." In particular, since 𝕊 is the monoidal unit in spectra instead of Hℤ, there's a ring map 𝕊⭢Hℤ, and we can "descend" along it. So 𝕊 is a base which is below ℤ!
4 cont'd: This sort of thing is somewhat wrapped up in ideas and writing of Jack Morava as well as Katia Consani and Alain Connes. See here for a little more about it: alainconnes.org/docs/gammasets…
5. There are also two major extensions of the theory of derived algebra: motivic homotopy theory, which (very roughly) does the whole story about spectra again but uses schemes instead of spaces and recovers the Weil cohomology theories (instead of the Eilenberg-Steenrod ones).
5 cont'd: And there's equivariant homotopy theory, which tries to redo the whole story of spectra but in the category of G-spaces, i.e. spaces with an action of a group. Both of these things have been and continue to be very active areas of research.
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