Let $C$ be a site. Localizing either the projective or injective model structure on simplicial presheaves at the Cech nerves of covers in $C$ gives a new model structure on simplicial presheaves which presents the $(\infty,1)$-category of simplicial sheaves on $C$. Passing to pointed simplicial presheaves, one can mimick the definition of sequential spectra and stable weak equivalences to obtain a stable model category which presents the stablization of the $(\infty,1)$-category of simplicial sheaves.

It seems to me that if one simply replaces "pointed simplicial set" by "pointed simplicial presheaf" in the construction of symmetric spectra. The notions of $S$-modules, symmetric smash product and mapping spectrum should proceed completely analogously to the classical case.

For the mapping space functor, one would use the cartesian closed structure on simplicial presheaves. The actions of the symmetric group would be given by embedding the symmetric group as a constant simplicial sheaf. The unit would be the simplicial sphere spectrum (again treated as a constant simplicial presheaf)

Question 1: Is there any obstruction to carrying this program out?

Question 2: If the answer to 1. is no, then is it obvious that the stable homotopy category one gets from symmetric spectra agrees with the stable homotopy category of simplicial sheaves?

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    $\begingroup$ Are you aware of Hovey's paper "Spectra and Symmetric Spectra in General Model Categories"? I think what you are sketching should work. Also, I'd tackle (2) via the universal property of stabilization $\endgroup$ – David White Aug 14 '16 at 15:43
  • $\begingroup$ @DavidWhite I'm not aware of that paper, but I'm sure I should be. I'll be sure to take a close look. Thanks! $\endgroup$ – Daniel Grady Aug 14 '16 at 16:44

Under some mild conditions, symmetric spectra in a monoidal model category admit a monoidal model structure with many nice properties (in particular, algebras over operads in symmetric spectra have a model structure of their own), see http://arxiv.org/abs/1410.5699.

In particular, as explained there, one can use the model category of pointed simplicial presheaves with the injective or projective model structure, producing a model structure on motivic symmetric spectra with nice properties.

The equivalence of this model structure with the nonsymmetric spectra (after discarding the monoidal structure on symmetric spectra) is a result due to Hovey, see Theorem 10.1 in Hovey's “Spectra and symmetric spectra in general model categories”.


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