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\begin{document}

\section{Road map, where we are}

Notations are not definitive. I've deleted the superscript $fib$ since
the double superscripting looks terrible after texing. 

We have the categories $\sE_G$ and $G\sE$ of finite $G$-sets.  We give
$\sE_G$ the bicategorical horizontal composition pairing using $\epz$, 
as is forced by studying tom Dieck splitting. 

We have the NU categories $\sF_G$ and $G\sF$ of orthogonal suspension 
spectra of finite $G$-sets. Write $\sF_G\un$ and $G\sF\un$ for adding 
units (notation will surely be changed later).

Unwritten NU version of 7.2 in the prequel, together with that result as 
written, should give:

\begin{thm}\mylabel{One} The categories $\sI\sS^{G\sF}$ (non-unital presheaves), 
$\sI\sS^{G\sF\un}$,
$\sI\sG^{\bK G\sE}$ (non-unital presheaves), and $\sI\sG^{\bK G\sE\un}$ are
Quillen equivalent.
\end{thm}

We can apply $\bN$ to these, or we can apply $\bM$; notations as in Mandell-May.

The same arguments applied to presheaves in $\sM$ should prove the analogue.

\begin{thm}\mylabel{Two} The categories $\sM^{\bN G\sF}$ (non-unital presheaves), 
$\sM^{\bN G\sF\un}$,
$\sM^{\bN\bK G\sE}$ (non-unital presheaves), and $\sM^{N\bK G\sE\un}$ are Quillen equivalent.
\end{thm}

If Dugger-Shipley works technically and the gods are good one can hope for 
direct comparisons among these. I'm pretty sure we know that $\sI\sS^{\BO}$ is Quillen 
equivalent to $\sM^{\BU}$. My guess is that this would want to go in two steps as follows,
but I wrote this before sketching out the Dugger-Shipley theory, so it should be adapted
to that.  This may be overoptimistic.

\begin{thm}\mylabel{Three} The four presheaf categories of \myref{Two} are Quillen 
equivalent respectively to four presheaf categories
$\sI\sS^{\bN^{\#}\bN G\sF}$ (non-unital presheaves), $\sI\sS^{\bN^{\#}\bN G\sF\un}$,
$\sI\sG^{\bN^{\#}\bN\bK G\sE}$ (non-unital presheaves), and $\sI\sG^{\bN^{\#}\bN\bK G\sE\un}$.
In turn these are Quillen equivalent equivalent respectively to the four presheaf
categories of \myref{One}.
\end{thm} 

I'm queazy about one thing in the unit/non-unit comparison.  In the EKMM version, we
have canonical quasi-identities $S \ltarr \mathbf{S} \rtarr G\sF(A,A)$.  Now the
equivalence between $G\sM$ and $\sM^{\sB}$ for sure talked about genuine unital 
sheaves. We pull back along $\sF\rtarr \sB$, and it is unclear to me how precisely
we account for the unital/non-unital dichotomy in asserting that $\sM^{\sB} ``='' \sM^{G\sF}$,
even though it feels almost tautological [Notations about when to write $G$ and when 
not are a mess, but I'm sure you understand.]  There is something about this that 
doesn't feel right.  Maybe a version of the quasi-equivalence trick will work here,
perhaps after transferring enrichement from $\sM$ to $\sI\sS$ to get a cofibrant unit 
object. Since we don't need the ``pointed'' condition, there might be a variant of this
trick that rigorously compares the thing we get by adjoining identities to the
version of $\sB$ that we are seeing. The $\sF$ in 7.5, fortuitously, should be the
$\sF$ here. Modulo fiddling, the quasi-identies should give something close enough
to the $\ze_d$ there, since nothing strict is needed. Then $\sB$ on the left, since
$\sF$ maps into $\sB$, and $G\sF\un$ on the right, since it maps onto $G\sF$, seems
to give just the right kind of structure. This is very rough, but at least gives an idea.

With less speculation and reliance on \cite{DS}, for sure we have 
\[ \sI\sS^{\sB} \htp G\sI\sS\htp G\sM \htp \sM^{\sB} \]
Barratt-Quillen surely should give the left equivalence (using NU presheaves) in
\[ \sI\sS^{\bK G\sE} \htp \sI\sS^{G\sF} \htp \sI\sS^{G\sF\un}. \]
It seems unimaginable that we cannot get equivalences between the last two
and their $\sM$-counterparts, and depending on queaziness we should have the 
second equivalence in   
\[ \sM^{G\sF} \htp \sM^{G\sF\un} \htp \sM^{\sB}. \]

I've completely lost the thread about keeping track of when we do or do not need
the $\sD(i,j)$ to be cofibrant. The only place I guess I'm sure it is important 
is in interpreting the Atiyah duality comparison. Note for that, by the way, that 
we have three rather than two things in sight, which is why I kept the middle row 
in the $3\times 3$ diagram on page 18: we have $F(\bA,\bB)$, $B\sma D\bA$, and 
$\bB\sma \bA$. We still have the strict unit in the middle level and it feels 
``almost'' cofibrant.

We have no reason to believe that $\bK G\sE$ is cofibrant in any sense stronger
than levelwise CW homotopy types (which we do have). However, we can rectify this.
This is maybe digressive, but here is how. We have the functor
$\bM\colon \sI\sS\rtarr \sM$, which is weakly equivalent to $\bN$
\cite{MM}. According to the five author paper, up to homotopy equivalence the
May machine is a Quillen left adjoint from $E_{\infty}$-spaces to LMS $\sS$. 
We can hit it with $\bF\colon \sS\rtarr \sM$. The functor $\bM$ is the top
composite in the diagram
\[ \xymatrix{
\sI\sS \ar[r]^-{U} \ar[dr]_{U}  & \sP[\bL] \ar[r]^-{L} \ar[d]^{U}
& \sS[\bL] \ar[r]^-{\bJ} \ar@{=}[dr]  \ar[d]^{U} & \sM \\
& \sP \ar[r]_-{L} & \sS \ar[r]_-{\bL} & \sS[\bL] \ar[u]_{\bJ} \\} \]
We have a natural weak equivalence $\epz\colon \bL U\rtarr \id$, the unit of an
adjunction, and the rest of the diagram commutes. The May machine $\bK'$ to $\sS$ is
related to the May machine $\bK$ to $\sI\sS$ by $\bK' = LU \bK$. Therefore we have a
natural weak equivalence $\epz\colon \bF\bK'\rtarr \bM\bK$. We can think of $\bF\bK'$
as the May machine to $\sM$, and viewed that way it is a Quillen left adjoint 
naturally equivalent to $\bM\bK$.  The functor $\bM$ is lax symmetric monoidal
\cite[7.11]{MM}.  There is a question about transporting that along $\epz$ that 
I haven't thought about but should be ok. Modulo that, this should give
\[  \sM^{\bF\bK' G\sE}\htp \sM^{\bM\bK G\sE} \]
showing that, when pushed into $\sM$ by the two obvious routes, we get eequivalent
presheaf categories with either interpretation. Using the weak equivalence $\bN\rtarr \bM$
and a Dugger-Shipley type argument should give
\[ \sM^{\bM\bK G\sE} \htp \sM^{\bN\bK G\sE}\htp \sI\sS^{\bK G\sE}. \]
The four displayed chains of equivalences give all the comparisons we should need.
More details to go, but the end is in sight.

\begin{thebibliography}{9}

\bibitem{DS}
D. Dugger and B. Shipley.
Enriched model categories and an application to homotopy endomorphism spectra. Theory and Application of Categories 18(2007), 400--439.

\bibitem{MM}
J. A. Mandell and J.P. May.
Equivariant orthogonal spectra and $S$-modules. 
Memoirs Amer. Math. Soc. Vol 159. 2002. 

\end{thebibliography}


\end{document} 