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][1] and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inverting $\mathbb{A}^1$-stable equivalences we obtain [Voevodsky's stable homotopy category][2] $\mathbf{SH}(S)$. 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 $\mathbf{Spt}(S)$? [1]: https://en.wikipedia.org/wiki/Noetherian_scheme [2]: https://ncatlab.org/nlab/show/motivic+homotopy+theory