Examples of non Quillen-equivalent model categories having equivalent homotopy categories

I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories.

The motivation is of course that I have two model categories and all the attempts to prove that they are Quillen equivalent have failed so far, but the homotopy categories are equivalent as category. I cannot find in this case any zig-zag of adjunctions between them. Reading such examples could help me to understand better the problem.

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Maybe people can help if you say a little bit about your categories. – Fernando Muro Jul 4 '13 at 8:13
Yes of course ; Should I post here or open a new thread ? I will give the definitions, and my attempts to find zig-zag of adjunctions, and why that does not work (so far). – Philippe Gaucher Jul 4 '13 at 8:59
I don't know, maybe in another thread since I guess you spect different kinds of answers. – Fernando Muro Jul 4 '13 at 9:00

The ones which usually come up to my mind as soon as I think of this:

• The categories of modules over $\mathbb Z/p^2$ and $\mathbb F_p[\epsilon]/(\epsilon^2)$, $p$ a prime integer. These rings are quasi-Frobenius, so their module categories have a model structure where cofibrations are monomorphisms, fibrations are epimorphisms and weak equivalences are homomorphisms which become isomorphisms in the stable module category, obtained from the module category by killing projective-injective objects. The homotopy category is this stable module category, which in both cases is the category of $\mathbb F_p$-vector spaces. This example is equivalent to Rasmus'.

• The category of DG-modules over $\mathbb F_p[v_n^{\pm1}]$, $|v_n|=2p^n-2$, $d(v_n)=0$, and the category of modules over Morava's $K(n)$.

• The model category of spectra localized at $K_{(p)}$-equivalences, where $K$ is complex $K$-theory and $p$ is an odd prime, and Franke's algebraic model category $C^{(T,N)}(\mathcal A)$ defined in http://www.math.uiuc.edu/K-theory/0139/. This doesn't happen for $p=2$ by results of Constanze Roitzheim.

In all these cases the model categories are stable and the homotopy categories are not only equivalent as categories but as triangulated categories.

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To the best of my knowledge these are the only known examples... Unless someone cares to update 'my knowledge'. – Dylan Wilson Jul 4 '13 at 10:54
@Dylan: There are variations on these examples. See for example Section 4.3 of Patchkoria's arxiv.org/pdf/1108.6309.pdf – Lennart Meier Jul 4 '13 at 12:52
... a paper which should be mentioned in the context of Franke's paper anyway. – Rasmus Bentmann Jul 4 '13 at 14:53

Daniel Dugger and Brooke Shipley given an example in their paper

A curious example of triangulated-equivalent model categories which are not Quillen equivalent

available here.

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