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Proposition 5.5 in $\mathbf{A}^1$-homotopy theory establishes the Brown's representability for the stable homotopy category $\mathcal{SH}_T(S)$, over a Noetherian scheme $S$, for a space $T$ of finite type over $S$.

Given a closed symmetric monoidal model category $(\mathcal{C},\mathcal{M})$, and an object $T\in \mathcal{C}$. Is there a reference proving when the stable homotopy category of $T$-symmetric spectra in $\mathcal{C}$, as in Spectra and symmetric spectra in general model categories, satisfies the Brown's representability. For instance, is the Brown's representability satisfied when $T$ is compact (for some notion of compactness) and when $\mathcal{M}$ is 'well-behaved' (e.g. cofibrantly generated, simplicial, proper,...)?

I recall reading such a statement, but I do not recall the source, or whether it was proven. So, I would be grateful if you could provide a reference.

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  • $\begingroup$ I am slightly confused: the Brown representability theorem depends only on the underlying homotopy category (or, if we want to be more refined, on the underlying ∞-category) and this does not depend on the model you choose. So the theorem holds in spectra, symmetric spectra, motivic functors... $\endgroup$ – Denis Nardin Feb 18 '17 at 14:35
  • $\begingroup$ @Denis I think the question is about categories unrelated to motivic theory. $\endgroup$ – Tyler Lawson Feb 18 '17 at 15:10
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    $\begingroup$ @DenisNardin: As Tyley guessed, the question is not about motivic categories. $(\mathcal{C},\mathcal{M})$ is an abstract model category, which I am happy to assume that is well-behaved in some sense. Another possible reformulation for the question would be "Was the result of Voevodsky, referred to in the question, generalised for (some) abstract stable homotopy categories of (symmetric) spectra?". $\endgroup$ – user24453 Feb 18 '17 at 15:24
  • $\begingroup$ @user24453 Sorry for misunderstanding your question. I'll let someone else answer in the case of symmetric spectra, but you might find interesting that there is such a statement for stable ∞-categories (theorem 1.4.1.2 in Lurie's Higher Algebra). $\endgroup$ – Denis Nardin Feb 18 '17 at 15:28
  • $\begingroup$ I believe the answer is "yes" in the following two situations (where I assume C is, say, combinatorial, and T will be compact and cofibrant and satisfy that condition about T^{\otimes 3} having homotopically trivial cyclic permutation): (1) if C has a set of compact, cogroup generators [and so already satisfies Brown representability], and (2) if T is a cogroup object. $\endgroup$ – Dylan Wilson Feb 18 '17 at 22:15

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