# “Stable model categories are categories of modules” - Clarification about a few things

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.

• When they say categories of modules they mean presheaf categories. (Right) modules over a ring $R$ correspond to presheaves on the one-object $\text{Ab}$-enriched category with endomorphisms $R$, and exactly analogous statements are true if abelian groups are replaced by spectra.There is no need to obscure what is going on here using the term "ringoid." It's like referring to a category as a monoidoid. – Qiaochu Yuan May 6 '15 at 6:17
• On (7): you can't compose a left Quillen equivalence with a right Quillen equivalence, so sometimes you only get a "chain" or "zigzag" of Quillen equivalences. – Zhen Lin May 6 '15 at 7:15
• Oops sorry for the delay, I was out of town, thanks Qiaochu, I was in a bit of a rush, I think I've been looking at "ring spectrums" from a very rigid point of view, Schwede's "An untitled book project about symmetric spectra" and Baker and Richter's "Structured ring spectra" seem like good introductory texts on the subject, going to give them a read. And Zhen, you mean (6)? You can compose a left Quillen equivalence with its respective right Quillen equivalence, or perhaps I'm misunderstanding what you're trying to say? – Samuel M May 11 '15 at 23:46