Holmstrom Stable model category

http://www.ncatlab.org/nlab/show/stable+model+category

A stable model category is a pointed model category whose homotopy category is triangulated, meaning that the suspension functor is an equivalence.

Examples: $Ch(R)$, for R a commutative ring. Also $Ch(B)$, for $B$ a commutative Hopf algebra over a field. S-modules. Symmetric spectra.

Non-examples: $Top_*$ and $Sset_*$

Can define closed monoidal triangulated category and show that the homotopy category of a stable monoidal model category is closed monoidal triangulated. See Hovey p. 178.

In a stable model category, homotopy pullback squares and homotopy pushout squares coincide.

arXiv:1204.5384 Stable left and right Bousfield localisations from arXiv Front: math.AT by David Barnes, Constanze Roitzheim We study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

nLab page on Stable model category