I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.


Let $S$ be a finite dimensional [Noetherian scheme](https://en.wikipedia.org/wiki/Noetherian_scheme) and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain [Voevodsky's stable homotopy category](https://ncatlab.org/nlab/show/motivic+homotopy+theory) $\mathbf{SH}(S)$. My question is:



> 
> Is there a model structure on $\mathbf{Spt}(S)$, having $\mathbf{SH}(S)$ as homotopy category, such that every object is fibrant? If so, could you provide a reference?
> 
> 
> 


For example, does the obvious candidate, given by the class of $\mathbb{A}^1$-stable equivalences as weak equivalences, surjective morphisms as fibrations, and cofibrations defined via the left lifting property, define a model structure on $\mathbf{Spt}(S)$? 



Model category structure on spectra

> 
> Is there a model structure on Spt(S), having SH(S) as homotopy category, such that every object is fibrant? If so, could you provide a reference?
> 
> 
> 


No, if the given model category of spectra Spt(S) (for which there are many
different, but Quillen equivalent, definitions) is not right proper,
because if all objects are fibrant, the model category is necessarily right proper.


Yes, if we are allowed to pick a specific Spt(S). According to Corollary 2.21 in a paper by Nikolaus (Algebraic models for higher categories), every cofibrantly generated model category C where all trivial cofibrations are monic is Quillen equivalent to a model category where all objects are fibrant.


Thus, one can take the projective or injective model structure on simplicial presheaves, localize it, pass to spectra (which gives one possible choice for Spt(S)),
and then apply the cited result to get a desired model category.


