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\begin{document}

\section{Sketch of how to think equivariantly}

Let $\sV$ be as usual, $G\sV$ the category of $G$-objects in $\sV$
with its induced symmetric monoidal structure, denoting $\sV_G(X,Y)$
the object in $G\sV$ given by $\sV(X,Y)$ with its conjugation action
by the discrete group $G$.  This will be a formal toy model.  We will
later want $\sV$ to be a good category of spectra, $G\sV$ to be a good
category of $G$-spectra enriched over $\sV$ (and not literally of the
form $G\sV$).   

We can apply our general theory to $G\sV$, viewed as enriched over $\sV$ 
via the hom objects $\sV_G(X,Y)^G$ in $\sV$. In fact, we can start over
with any $\sV$-enriched $\sM$ and form $G\sM$ and $\sM_G$ analogously,
enriched over $\sV$ and $G\sV$ respectively.  Let $G\sD[full]$ 
be an appropriate full subcategory of $G\sM$, enriched over 
$\sV$ via the $\sM_G(X,Y)^G$  and let $\sD_G[full]$ denote the 
corresponding full subcategory of $\sM_G$, enriched over $G\sV$ via the 
hom objects $\sM_G(X,Y)$.  We give meaning to the following diagrammatic 
road map, which is written on the level of ordinary model categories.

\[ \xymatrix{
G\sM_0 \ar@<1ex>[r]^-{\bT} 
& \ar@<1ex>[l]^-{\bU} \sP(G\sD[full], \sV)_0  \ar@<1ex>[r]^-{{\epz}^*}  
& \ar@<1ex>[l]^{(-)^G} \sP_G(\sD_G[full], G\sV)_0 \ar@<-1ex>[d]_-{\de^*}\\
\sP(G\sD[disc], \sV_0)\ar@<1ex>[r]^-{{\epz}^*} 
& \ar@<1ex>[l]^-{(-)^G} \sP_G(\sD_G[disc], G\sV_0) 
& \sP_G(S_G[\sD_G[disc]], G\sV)_0 \ar@<-1ex>[u]_-{\de^{\#}} \ar[l]_-{\iso} \\}\]

\begin{step}\mylabel{one} Prove an enriched Quillen equivalence between 
$G\sM$ and
$\sP(G\sD[full],\sV)$.
\end{step}

This we already know how to do: our general theory applies.

\begin{step}\mylabel{two} Prove a Quillen equivalence (probably enriched 
over $\sV$ but that's gravy) between the underlying categories of 
$\sP(G\sD[full], \sV)$ and $\sP_G(\sD_G[full], G\sV)$.
\end{step}

If $\epz\colon G\rtarr e$, then $\epz^*$ is pullback and assigns 
trivial action of $G$ to objects of $\sV$.  Its right adjoint is
$(-)^G$.  We use $(-)^G$ to create a model structure on 
$\sP_G(\sD_G[full], G\sV)_0$.  It perhaps requires special good
properties of the fixed point functor, but this too we know how
to do in principle. It really depends only on $\sV$, not $\sM$: we
could start with any pair $(\sD_G,G\sD)$. Since $\et\colon X\rtarr (\epz^*X)^G$ is the identity in our current formal context, this will certainly
be a Quillen equivalence. NO: $\epz^*$ is not defined on presheaves!
However, $(-)^G$ is fine. Alternative prolongation construction?

\begin{step}\mylabel{three} Find a discrete $G$-category $\sD_G[disc]$ with $G$-fixed
point category $G\sD[disc]$ together with a $G$-equivalence 
$\sD_G[disc] \rtarr \sD_G[full]$.  Then use it to construct an
enriched Quillen equivalence between $\sP_G(\sD_G[full], G\sV)$
and $\sP_G(S_G[\sD[disc]], G\sV)$.
\end{step}

The first part should be the hard part, and in the $G$-spectrum case
it is the thing that we have been looking for in terms of transfers on
the one hand and our understanding of $F_G(X,Y)$ for orbit $G$-spectra
$X$ and $Y$, or really, for any $G$-spectra $\SI_G\SI^{\infty}(K)_+$, 
where $K$ is a $G$-set, on the other hand.  Obviously, in general the
answer depends on the initial choice of $\sM$.  In our $G$-spectrum
case, it is ``only'' a case of finding a good description of $(\sD_G,G\sD)$
and a map $\de$.  We know that it is there morally from our understanding
of the $G$-homotopy types of the $F_G(\bar{K},\bar{L})$, where 
$\bar{K} = \SI^{\infty}_G(K_+) = S_G[K_+]$ for a finite $G$-set $K$:  
they are $S_G[(K\times L)_+]$.

\begin{step}\mylabel{four} Prove that $\sP_G(S_G[\sD_G[disc]], G\sV)_0$ is 
isomorphic as a model category to $\sP_G(\sD_G[disc], G\sV_0)$.
\end{step}

This should be the same as we saw in general, in the nonequivariant
context.  The notation $\sP_G$ is really more for emphasis than for
mathematics.  Again, this should be a general argument, independent 
of $\sM$. 

\begin{step}\mylabel{five} Prove a Quillen equivalence (no enrichment) 
between $\sP_G(\sD_G[disc], G\sV_0)$ and $\sP(G\sD[disc], \sV_0)$.
\end{step}

This should be essentially the same as \myref{two}, but perhaps simpler,
and again general, independent of $\sM$. As a test of sanity, we should 
make sure that this program works trivially when
$\sM=\sV = \sU$ and $G\sD = G\sO$; $\sO_G$ will have orbits and set 
maps between them, with $G$ acting by conjugation.   

\end{document}
\section{Discrete group actions in $\sV$-categories}\label{DGV}

UNEDITED.

There are two quite different generalized versions of \myref{Gspaces}.
There we worked with actions by a group object in the 
cartesian monoidal category $\sU$.  Instead, we can work either with 
discrete groups and general $\sV$-categories $\sM$ or with group 
objects in general cartesian monoidal categories $\sV$.  We consider 
the first of these contexts here. Thus let $G$ be a discrete group, 
let $\sV$ be a symmetric monoidal model category as specified in \S1, 
and let $\sM$ be a cofibrantly generated $\sV$-model category.
 
Recall \myref{Good2}. We agree to view $G$ as a category with a 
single object, $*$ say, and maps the elements of $G$.  We think of
$G$ as a discrete diagram category, and then $S[G]$ as defined in
\myref{Good2} is the $\sV$ analogue of the group ring of $G$.   
When $\sV$ is the category of $k$-modules or of chain complexes 
over $k$, it is precisely the group ring $k[G]$, regarded as a 
$\sV$-category with a single object.  

The underlying category of the $\sV$-category $G\sM = \sP(S[G],\sM)$ 
is the category $\sP(G,\sM_0)$ and that is just the 
ordinary category $G\sM_0$ of $G$-objects in $\sM$. For $X,Y\in G\sM$, 
the mapping hom 
object $G\sM(X,Y)$ in $\sV$ is defined formally as an equalizer in $\sV$.
For $X\in G\sM$ and $V\in \sV$, the tensor $X\odot V$ and cotensor 
$F(V,X)$ in $G\sM$ are just $X\otimes V$ and $\sV(V,X)$ with the actions 
of $G$ induced by the action of $G$ on $X$.  Evaluation at the single 
object $*$ forgets the $G$-action, and its left adjoint $F_*$ sends an 
object $V\in \sV$ to the free $G$-object $S[G]\odot V$ in $G\sM$.  
Since $S[G]$ is a Quillen $\sV$-category, \myref{level2} applies.

\begin{prop} 
$G\sM$ is a cofibrantly generated and proper $\sV$-model category 
with weak equivalences and fibrations obtained by forgetting the 
$G$-action.  The sets $S[G]\odot I_{\sM}$ and $S[G]\odot J_{\sM}$ 
are the generating cofibrations and generating acyclic cofibrations.
\end{prop}

This is the naive model structure, rather than a truly equivariant one.
For that, we need orbit categories, and there are actually two quite different
types in sight.  First, for a family $\sF$ of subgroups of $G$, we have the discrete category $\sO_{\sF}$ of orbits $G/H$ with $H\in \sF$ and $G$-maps between
them, and \myref{Good2} gives a $\sV$-category $S[\sO_{\sF}]$.  With this
choice, we have
\[ \sP(\sO_{\sF},\sM)_0\iso \sP(S[\sO_{\sF}],\sM)_0. \]

Second, we can construct a full $\sV$-subcategory $\sD_{\sF}$
of $G\sV$ whose objects are the $S[G/H]$ for $H\in\sF$.  The
homs now are 
\[  \sD_{\sF}(S[G/H],S[G/K]) = \sV(S[G/H],S[G/K]) \]
These two $\sV$-categories can be identified when $\sV$ is cartesian
monoidal, but not in general otherwise.  For example, $\sV_0(*,*)$
is a point, where $*$ is the terminal object, and this is very
different from $\sV_0(S,S)$ when $S$ is the unit object for a 
different monoidal structure $\otimes$ on $\sV$.  

Observe that $S[G/H]$ can be identified with the
evident categorical tensor product $S[G]\otimes_{S[H]}*$, where $*$ 
is viewed as a left $H$-object.  More generally, for an $H$-object $W$ 
in $\sV$, we define a $G$-object $S[G]\otimes_{S[H]} W$ in $\sV$. 
We also define a $G$-object $\sV_{S[H]}(S[G],W)$ in $\sV$ as the 
equalizer of the pair of maps 
$$\sV(S[G],W)\rtarr \sV(S[H]\otimes S[G],W)$$
obtained by interpreting $H$-equivariance, $f(hg) = hf(g)$,
in terms of internal hom objects; we use the right action of $G$ 
on itself to obtain an induced left action of $G$ on the equalizer.
The inclusion $\io\colon H\rtarr G$ induces a $\sV$-functor
$\io^*\colon G\sV\rtarr H\sV$.

\begin{lem}  There are $\sV$-adjunctions
\[ G\sV(S[G]\otimes_{S[H]}Y,X)\iso H\sV(Y,\io^*X) 
\ \ \text{and} \ \ H\sV(\io^*X,Y)\iso G\sV(X,\sV_{S[H]}(S[G],Y),\]
where $X\in G\sV$ and $Y\in H\sV$.
\end{lem}

For $Y\in H\sV$, such as $Y=\io^*X$, we also have the orbit objects 
$Y/H$ and fixed point objects $Y^H$ in $\sV$.  These are the colimit
and limit of $Y$ regarded as a functor on $H$.  Equivalently,
in terms of the terminal object $*\in H\sV$, these are 
\[ Y/H = Y\otimes_{S[H]} * \ \ \ \text{and} \ \ \ Y^H= \sV_{S[H]}(*, Y). \] 
These functors actually take values in $WH\sV$, where $WH = N_GH/H$, but
we ignore that. The trivial homomorphism $\epz\colon H\rtarr \{e\}$
induces a $\sV$-functor $\epz^*\colon \sV\rtarr H\sV$.  

\begin{lem} There are $V$-adjunctions
\[ H\sV(Y,\epz^*V)\iso \sV(Y/H,V) \ \ \text{and} \ \  
H\sV(\epz^*V, Y)\iso \sV(V,Y^H), \]
where $Y\in H\sV$ and $V\in\sV$. 
\end{lem}

Observing that we have a natural isomorphism,
\[ S[G]\otimes_{S[H]}\epz^*V \iso S[G/H]\odot V \]
and omitting $\io^*$ from the notation, we can compose adjunctions to obtain
\begin{equation}\label{keyG}
 G\sV(S[G/H]\odot V,X)\iso \sV(V, X^H).
\end{equation}
We are particularly interested in the case $V=S$.  

Again, $S[\sO_{\sF}]$ is a Quillen $\sV$-category, 
and \myref{level2} applies. We write $X_{H}$ for the levels
$X_{G/H}$ of a prespectrum $X\in \sP(S[\sO_{\sF}],\sV)$.

\begin{prop} $\sP(\sO_{\sF},\sV)$ is a cofibrantly generated and proper 
$\sV$-model category.  The sets $S[G/H]\odot I$ and $S[G/H]\odot J$,
$H\in\sF$, are the generating cofibrations and generating acyclic 
cofibrations, and for all $S[\sO_{\sF}]$-cofibrations $f\colon X\rtarr Y$, 
$f_{H}\colon X_{H}\rtarr Y_{H}$ is a cofibration in $\sV$.
\end{prop}

We now ask \myref{quest1} with $\sM=G\sV$ and $\sD = S[\sO_{\sF}]$,
or we can ask \myref{quest3} with $\sD=\sD_{\sF}$.  We focus on the
first question first. When $\sF = \{e\}$, $S[\sO_{e}] = S[G]$.  Since $G\sV$ and the 
presheaf category $\sP(S[\sO_{\sF},\sV)$ are the same, there is 
nothing to say in
In general, (\ref{keyG}) implies that the $\sD_{\sF}$-equivalences 
and $\sD_{sF}$-fibrations in $G\sV$ specified in \myref{newmod} are 
the $G$-maps $f\colon X\rtarr Y$ such that $f^H\colon X^H\rtarr Y^H$ 
is a weak equivalence or fibration in $\sV$ for all $H\in\sF$; these 
are usually called $\sF$-equivalences and $\sF$-fibrations.  The case 
$\sF = \sA\ell\ell$ of all subgroups of $G$ is of greatest interest, 
and then these are the weak $G$-equivalences and the Serre
$G$-fibrations in $G\sV$.

To have a $\sD$-model structure on $G\sV$, we must show that $\sD$
is very good. 

\section{Group actions in cartesian monoidal categories}\label{GCM}

We here consider group objects in a general cartesian model 
category $\sV$.  Thus the product in $\sV$ is the 
categorical product, and otherwise the assumptions on $\sV$
are as in \S1 and \S2.  

\[ G\sV(X,\sV_{S[H]}(S[G],\epz^*V)\iso \sV((\io^*X)/H,\epz^*V).\]
Now, for the first time using that $G$ is a group and not just a monoid,
we note that $G\sV$ is a closed symmetric monoidal $\sV$-category. 
For $X,X'\in G\sV$, the tensor $X\otimes X'$ in $\sV$ inherits the 
diagonal $G$-action, and the internal hom $\sV(X,X')$ inherits the
conjugation $G$-action.  This can also be interpreted as giving an
enrichment of $G\sV$.  That is, $G\sV$ is the 
