Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where $\mathrm{sSet}_*$ is the Quillen model category of pointed simplicial sets.

With this structure one can define suspension and loop functors on $\mathrm{Ho}(\mathcal C)$. The suspension $\Sigma$ is the (derived) smash product with the simplicial $\mathbf S^1$ and the loop functor is its right adjoint. By definition, $\mathcal C$ is called *stable* iff $\Sigma$ is an autoequivalence of $\mathrm{Ho}(\mathcal C)$.

In case $\mathcal C$ is not necessarily stable, I'm looking for canonical ways to stabilize $\mathcal C$. In case $\mathcal C$ fulfills extra axioms, e.g. it is left proper and combinatorial, one can find by [1] a simplicial model structure on the simplicial objects $\mathrm s \mathcal C$ of $\mathcal C$ which is Quillen equivalent to $\mathcal C$ itself. So for left proper and combinatorial model categories $\mathcal C$, one may assume that they are already simplicial so that the suspension is already defined on the level of the model category. Then one can use [2] or [3] to consider (symmetric) spectrum objects in $\mathcal C$ with respect to the suspension and gets a canonical stable model category, where the model structure is the stable one.

Now my question is as follows: Is there a canonical way to stabilize a model category without going through simplicial enrichments (and which may work with any closed model category)? Bonus questions are: Are there better or more canonical references than the two I gave? Does it make sense just to stabilize the homotopy category $\mathrm{Ho}(\mathcal C)$, e.g. just to consider spectra on the homotopy level?

[1] Dugger: Replacing Model Categories with Simplicial Ones

[2] Hovey: Spectra and symmetric spectra in general model categories

[3] Schwede: Spectra in model categories and applications to the algebraic cotangent complex

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