Check our Theorem 2.10 against Hirschhorn's Theorem 11.6.1 (and 11.6.23).  
He has no hypotheses.  Can our result be improved?

\section{Questions, reminders, more to do}

SHOULD PUT IN  maybe in \S2 (or \S4)  already: 
A $\sV$-equivalence $\sD\rtarr \sE$ induces a Quillen
equivalence between $\sP(\sD,\sM)$ and $\sP(\sE,\sM)$. 
See SS, A.1.1.(iii) or MMSS, 12.7.

Maybe add reference to Mandell-May, prolongation: the construcution of $\bT$
is a prolongation of $\de$ along $\bD$, where $\bU$ corresponds to the
give there.  

Maybe add remark on combinatorial model structures (Beke, Dugger,
Dugger-Shipley) and avoidance of preassigned J.

Problem type: $\sD$ versus $\text{Ho}\sD$.  When can we shrink things down?

Equivariant categories of spectra.  We are interested in examples of 
the same general nature as \myref{Gspaces}, such as categories of $G$-spectra. 
Enrich via $i_*$ and smash of G-spectra. Bertrand's notes have worked this out. 

Equivariant motivic homotopy theory?  

Note: point set level category is big.  Compare LMS p. 284.  Homotopy
level small.   By adjunction, seeing $G\sS\!p(G/H,G/K) = Q_G(G/K)^H$,
where $G/H$ is short for $\SI^{\infty}_G G/H_+$; note that 
$\SI^{\infty}_G = i_*\SI^{\infty}_G$ comparing naive and genuine
suspension. 

Burnside category: for finite G, embed $\sB_G$ in complete universe
version.  Induces a Quillen equivalence of diagram categories? 

Kan extension along $\de\colon \sD\rtarr \pi_0\sD$?  Get discrete
diagram from continuous one?  When is this a Quillen equivalence?

Parametrized spectra: seriously interesting.  The difference between
model theoretic generating sets (cofibers of generating cofibrations)
and triangulated homotopy category generating sets is enormous, and the 
latter are sufficient from the diagram point of view.  This is strange, 
since these generators do not see the topology, only the points, of B, 
even just one point in each component.  Fibrant approximation is the 
key to understanding this.
