MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 10 down vote accepted

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 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.

share|cite|improve this answer
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 – 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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.