I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things:
1 - When they say that stable model categories are categories of modules, are they talking about "traditional" modules as defined in http://en.wikipedia.org/wiki/Module_(mathematics)? So $End(P)$ in theorem 3.1.1 is a ring (not a ringoid), and $mod-End(P)$ is the category of modules over a ring in the traditional way?
2 - In theorem 3.1.1, is $End(P) = Hom_{Sp(\mathcal{C})}(\Sigma_f^{\infty}P, \Sigma_f^{\infty}P)$ (from definition 3.7.5)?
3 - In the last paragraph of page 15, $Hom_{Sp(\mathcal{C})}(X, Y)$ is defined as "the equalizer of the two maps $Hom_{\Sigma}(X,Y) \rightarrow Hom_{\Sigma}(S \otimes X,Y)$". Which two maps? They are not defined.
4 - In what way is $Hom_{Sp(\mathcal{C})}(\Sigma_f^{\infty}P, \Sigma_f^{\infty}P)$ a ring? Am I supposed to look at it as an endomorphism ring of some sort?
5 - What is a "chain of simplicial Quillen equivalences"? I went to nlab and I found the entry on enriched Quillen adjunction/equivalence between enriched categories $C$ and $D$, from what I sort of gathered it seems to be a Quillen adjunction/equivalence between the "underlying categories" $C_0$ and $D_0$, but I'm interested in finding a Quillen equivalence between a stable model category $\mathcal{C}$ and $mod-End(P)$ (theorem 3.1.1) not between $\mathcal{C}_0$ and $mod-End(P)_0$?
6 - Why are they "chains"?
7 - In this paper "map" is the simplicial set of morphisms in a simplicial category and $hom_C$ is the set of morphisms in a category $C$, if in the last paragraph of page 15 $Sp(\mathcal{C})$ is a simplicial category, isn't $map(X,sh_nY) = hom_{Sp(\mathcal{C})}(X,sh_nY)$ automatically? or is this "map" in this paragraph referring to a different set of morphisms? Just want to make sure.