Holmstrom Symmetric spectra

The original article, by Hovey, Shipley and Smith, is here.

Stefan Schwede has an interesting online book project

Jardine on presheaves of symmetric spectra: “This paper shows that there is a proper closed simplicial model category on the category of presheaves of symmetric spectra on an arbitrary Grothendieck site, and that the resulting homotopy category is equivalent to the stable category of presheaves of spectra.”

Jardine on A1-local symmetric spectra: http://www.math.uiuc.edu/K-theory/0350. This article is replaced by Motivic symmetric spectra: “The paper demonstrates the existence of a theory of symmetric spectra for the motivic stable category. The main results imply the existence of a categorical model for the motivic stable category which has an internal symmetric monoidal smash product. More explicitly, it is shown that there is a proper closed simplicial model category structure for the category of symmetric T-spectra, suitably defined, on the smooth Nisnevich site of a noetherian scheme of finite type. The weak equivalences for this structure are stable equivalences, defined by analogy with the definitions given by Hovey, Shipley and Smith for ordinary symmetric spectra and by Jardine for presheaves of symmetric spectra, except that one suspends by the Morel-Voevodsky object T, and the underlying unstable category is the motivic closed model structure for simplicial presheaves on the Nisnevich site. The homotopy category obtained from the category of symmetric T-spectra is equivalent to the motivic stable category. ”

Hovey on Spectra and symmetric spectra in general model categories

Sniggy gave a talk I think where he defined the model structure via an inclusion into a cat $Fct(\Sigma, sSet_*)$ for a certain small cat $\Sigma$. Also a description of the symmetric monoidal structure in terms of a left Kan extension.

arXiv:1108.3509 Positive model structures for abstract symmetric spectra from arXiv Front: math.AT by Sergey Gorchinskiy, Vladimir Guletskii We prove the existence of a suitable “positive” model structure for symmetric spectra over an abstract simplicial monoidal model category. This allows to generalize the theorem due to Elmendorf, Kriz, Mandell and May saying that the $n$-th symmetric power of a positively cofibrant topological spectrum is stably equivalent to the $n$-th homotopy symmetric power of that spectrum, see [EKMM], III, 5.1, and [MMSS], 15.5. As a consequence, we also prove the existence of left derive symmetric powers for abstract symmetric spectra. The results are general enough to be applicable to the Morel-Voevodsky’s motivic symmetric spectra of schemes over a field.

nLab page on Symmetric spectra