What's Aut(E_n) where E_n is the E_n-operad? Could it just be O(n)?

So Rognes defines, for a finite group $G$, a $G$-Galois extension of $E$-local $E_\infty$ ring spectra (but I think the definition makes sense at least in any symmetric monoidal stable presentable $\infty$-category) to be a homomorphism $A \to B$ which (1) induces an equivalence $A \to B^{hG}$ and (2) induces an equivalence $B \wedge_A B \to \prod_G B$. I'm told this corresponds to a classical definition of Galois extension, but I've never seen this classical definition before.

(1) seems natural but (2) looks mysterious.

Like, does this still coincide with the condition of being finite etale, when you plug in those definitions from HA or SAG?

I suppose I don't actually know classically why finite + normal + separable is the same as finite etale. Maybe I should start there.

@TimCampion The key word here is "torsor"

More concretely, a G-action on a set X is free iff the map G×X→X×_{X/G}X sending $(g,x)$ to $(x,gx)$ is an isomorphism

Hm

@SaalHardali This is an open question. There is a rational computation due to Fresse and Willwacher that suggests that it's not O(n).

I will have to work through that as an exercise.

It's very reminiscent of the argument that if $C$ is a Waldhausen category where every cofibration splits, then the $K$-theory $K(C)$ is the same as the group completion of $C$ under coproduct.

@TimCampion This is a good account of the classical theory

Although maybe you don't want to read all of it.

:)

@TimCampion I don't quite see the connection

Remember that Rognes here is generalizing Galois theory for rings, not only the case of Galois theory for fields

Well, the argument of the Waldhausen thing comes down to showing that if you look at a co-slice category $C_{X/}$, then there is an isomorphism $A \cup_X A \to A \vee (A/X)$, natural in $A \in C_{X/}$.

Which seems at least formally very similar -- and the hypothesis that every cofibration splits is formally similar to the freeness hypothesis

Hrmm.. I guess. We are working with a much stronger condition here though

how so?

Well, for me a split Waldhausen category is very "squishy", it's not a very rigid kind of structure. While a Galois extension is extremely rigid. Said differently, the Galois extension is completely controlled by the Galois group, but the split Waldhausen category is not completely controlled by the K-theory

fair

Maybe more helpful: when you do Galois theory for fields, one of the first things you prove is a lemma by Artin that says that $[F:F^G]=\#G$. This is sort of the analogous condition for rings, only you cannot get it for free and you have to impose it

But it is equivalent to being finite etale, right

?

Well, you also need the condition that A^G→A is faithfully flat

Again, something that's automatic for fields, not anymore for rings

For some reason Rognes splits this in a separate condition

Ok -- so Rognes Galois + faithfully flat = finite etale

And Galois is stronger than finite étale. It is finite étale + some kind of transitivity condition

shoot

Think about covering spaces: not all finite covering spaces (i.e. finite étale maps) are Galois (i.e. the monodromy acts transitively on each fiber)

it's weird because in topology we tend not to impose that as a condition

Neither do people in algebraic geometry

Galois is a special case, although a very important one and worth of study

In any case, the proof that a G-torsor is finite étale is ridicolously simple

hm

so the different axiomatizations don't have their axioms corresponding precisely

Really, you should think "principal $G$-bundle" rather than covering space here

It's weird to me that Lurie's notion of "etale" kind of implies "discrete fibers", whereas in $\infty$-topos theory, "etale" (also Lurie's notion!) doesn't have this implication

Lurie's notion of étale is not really intrinsic. It uses a lot the t-structure on Mod_R

That's why it makes sense only for connective things (i.e. ring spectra that do have a t-structure on Mod_R)

Is there a definition that makes sense without a t-structure which specializes to it?

Rognes here is trying to study something more general, especially something that will work for even-periodic things

ah!

@TimCampion Specializing to what?

Not all Galois extensions are étale in the sense of Lurie.

For a simple example, KO→KU

I'm wondering if there's a notion of "etale functor between symmetric monoidal stable presentable $\infty$-categories" which, in the case where the categories are $Mod_R$ and $Mod_S$, specializes to the base-change adjunction for a map $R \to S$

probably not -- that's what you mean when you say "not intrinsic"

Yeah, I doubt it.

I also wonder if there's a meaningful notion of "$G$-Galois cover" when $G$ is not discrete (or profinite)

Well, you can define it for every commutative Hopf algebra, not only those coming from (pro)finite groups

That's probably better than having a definition for topological groups

since a Hopf algebra is like an internal group in the opposite category.

Yeah, indeed. That's why it works :)

Do you happen to know an interesting example of one of these?

Topologically or classically?

Topologically there's S→MU, which is a S[BU]-torsor

There's the problem that it's not faithful

Of course classically there's plenty of principal bundles for algebraic groups

cool!

i guess i was thinking topologically, so that's perfect -- but I suppose it wouldn't hurt to know a classical example too

Dunno, $\mathbb{A}^{n+1}\smallsetminus\{0\}→\mathbb{P}^n$ is a $\mathbb{G}_m$-torsor

@GeoffroyHorel Interesting! thx

Regarding the earlier discussion about Galois covers as opposed to general covers: the ones where the group acts transitively are (one hopes) cofinal in the system of all covers. This is true in the topological case, Lubkin calls these regular covers -- and in the scheme-theoretic etale case, where finite etale maps are dominated by Galois ones.

@JoeBerner It's true in every Galois category, to be precise :)

(which is just a fancy way of saying that normal subgroups are cofinal in the poset of all closed subgroup of a profinite group)