i'm most skeptical of my identification $Sp^{hG} \simeq Mack(Burn(GFin^{free}))$, and now i'm thinking that actually we only have $Sp^{hG} \simeq Fun^\oplus((GFin^{free})^{op},Sp)$ -- is that the entire issue, the additive transfers?
hmm, let me just try unwinding this. you're saying that for a map $S \to T$ in $Fin$, we need $S \to (S \times G/e)^G \to (T \times G/e)^G$ and $S \to T \to (T \times G/e)^G$ to define a pullback square?
I didn't really read the above discussion but, in case someone is asking, the right adjoint to the inclusion of geometric spectra is, I believe, the 'cogeometric fixed points' given by F(\tilde{EP}, -)^G ? It's not used very often for some reason
also omg... I just checked three different references and found three different conventions for whether i or j should be for open or closed embeddings, and it's making me insane. left and right is confusing enough without also having to keep track of this
I'm this close to calling open embeddings "Open" and closed embeddings "Closed" but I think that would upset everyone... including myself.
The open embedding is j. Please do not use any other convention...
But be careful that the role of the open and closed embedding swap between constructible sheaves and quasi-coherent sheaves (that might be the reason for differing conventions)
@Drew A Bousfield localization $f:X\to Y$ is characterized by (i) $Y$ is $B$-local, i.e., there are no maps of $B$-acyclics to $Y$, and (ii) the fiber of $f$ is $B$-acylic. Just apply this directly to $X\to B\wedge X$ when $B$ is smashing.
BTW, for my original question: assume $B$ is a unital $A$-module which is smashing (whence $B$ is an $A$-algebra) and dualizable. Let $F=$ fiber of $A\to B$, which will also be dualizable since $A$ and $B$ are. Then the dual of $F\to A$ is a map $A\to F^*$, which will be also be smashing, whence $F^*$ is an $A$-algebra. ....
Then $F^*\otimes_A B=0$, because this is the same as the function $A$-module $[F,B]$, which is contractible since any $F\otimes X\to B$ factors through $B\otimes F\otimes X=0\otimes X$. Thus $A\to B\times F^*$ is smashing, and is also conservative by a similar argument, so is an equivalence by @DenisNardin's argument.
Sorry to ask such a simple question, but does anyone have a reference for the spectra X_i, defined in page 2 of this pdfs.semanticscholar.org/1251/…
They are defined as the Thom spaces of the maps f_i:\Omega S^i\to BO, which are given as the loops of generators of the homotopy groups of B^2O. I ask as I am unable to verify the properties listed in section 2.