@CPM yup, that sounds right to me too; i'm reading a paper that seems to totally ignore that fact (and works really hard as a result!), so i wanted to double-check to make sure i wasn't crazy
A friend of mine gave me a book by Alain Badiou today where he apparently attempts to construct a sort of philosophy of philosophy using some kind of set theory.
Moreover, any $A\ltimes M$-module is necessarily an $A$-module, so going up a level basically means restricting to a certain class of $A$-modules. I wonder if this is written down anywhere.
As in, what the classification of such $A$-modules is.
Actually, I believe both Austria and Australia are pro axiom of global choice - there are radical fringe finitist movements, but they are pretty harmless.
I once went to a talk entitled "2-groups and 2-bundles" and I really wanted the speaker to go to the board, write "SO(3), quaternions" and "tangent bundle of a sphere, trivial bundle on any space " and then leave
@EricPeterson: By the way Eric here is a fun curiosity about Thom spaces/spectra I recently discovered from LMS. Take the Thom space $Tf$ of an $S^n$ fibration over a connected base space $X$. Then $\pi_n Tf\cong \mathbb{Z}$ or $\mathbb{Z}/2$ moreover the first option happens if and only if the classifying map $X\rightarrow BhAut(S^n)$ lifts to $BShAut(S^n)$.
@EricPeterson: This implies Thom spectra coming from connected spaces can only have $\pi_0$ equal to $\mathbb{Z}$ or $\mathbb{Z}/2$ and this is determined by whether or not the classifying map lifts to $BSL_1S$. In particularly if $f$ is a loop map and we take the associated Thom ring spectrum then either $\pi_0$ is $\mathbb{Z}$ or the homotopy ring is a $\mathbb{Z}/2$ algebra (e.g. MO).
An easy consequence of this is is that odd Eilenberg-MacLane spectra are not Thom spectra.
That's a cool fact that is also used in Fridolin Roth's thesis to determine that HZ-oriented Thom spectra are always Hopf-Galois extensions, or something like that.
@JustinNoel that is nice. :) i know that there's a competing fact, but i forget how it goes: by replacing BGL_1 S with BGL_1 S_(p), you can produce either HZ_(p) or HZ/p as a thom spectrum over some base which is morally similar to Loops^2 S^3. does something change dramatically enough that you actually can produce HZ/p from BGL_1 S_(p), or can you only get the much more mundane and believable HZ_(p)?
