When is a thick subcategory the preimage of a weak Serre class under a homological functor?

Question

Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $\pi^{-1}(\mathcal C) \subseteq \mathcal T$ be the full subcategory of objects $T$ such that $\pi(\Sigma^n T) \in \mathcal C$ for all $n \in \mathbb Z$. Then $\pi^{-1}(\mathcal C)$ is a thick subcategory of $\mathcal T$ ("by construction" -- this is kind of the whole point of weak Serre classes!)

Question: Suppose that $\mathcal P \subseteq \mathcal T$ is a thick subcategory of a stable / triangulated category $\mathcal T$. Under what conditions does there exist an abelian category $\mathcal A$, a homological functor $\pi : \mathcal T \to \mathcal A$, and a weak Serre class $\mathcal C \subseteq \mathcal A$ such that $\mathcal P = \pi^{-1}(\mathcal C)$?

I'm also interested in the version of this question where $\mathcal T$ is compactly-generated, $\mathcal A$ is required to be Grothendieck, and $\mathcal \pi$ is required to commute with filtered colimits.

EDIT: I realize that when $\mathcal P$ is closed under colimits, then it's fruitful to think of this question in terms of the localization $\mathcal T / \mathcal P$. But, as I should have specified, this question is motivated primarily by the case where $\mathcal P$ is a thick subcategory which is not closed under coproducts -- in fact, I'd be happy to restrict to the case where $\mathcal P$ is small. For example, consider the case where $\mathcal T$ is the category of spectra and $\mathcal P$ is the full subcategory of finite-type spectra. Then $\mathcal P = \pi^{-1}(\mathcal F)$, where $\pi : \mathcal T \to Ab$ takes homotopy groups, and $\mathcal F \subset Ab$ is the full subcategory of finitely-generated abelian groups. I'm wondering to what extent one can understand small thick subcategories in terms of homological functors.

Answer 1

"Almost certainly" (as far as I remember, one needs either a set of generators for $\mathcal P$ or the Vopenka principle; see Theorem 7.2.1 of Krause's https://arxiv.org/abs/0806.1324) if $\mathcal{P}$ is closed with respect to $\mathcal{T}$-coproducts then the Verdier quotient $\mathcal{T}/\mathcal{P}$ is locally small and well generated. Thus if $L:\mathcal{T} \to \mathcal{T}/\mathcal{P}$ is the localization functor then you can compose it

(i) either with the collection of functors corepresentable by the corresponding generators. This will give a collection of functors into abelian groups whose kernel is $\mathcal{P}$. Unfortunately, these functors do not have to respect coproducts.

(ii) or with the functor into the category $\hat{\mathcal{S}}$ of coherent functors, where $\mathcal{S}$ is the $\coprod$-closure in $\mathcal{T}/\mathcal{P}$ of a perfect set of generators $\mathcal{S}_0$; see Krause's https://eudml.org/doc/49219. You will obtain a functor that respect coproducts and whose target is AB4.

I don't know how to respect coproducts and make the target Grothendieck abelian simultaneously, sorry. Possibly, Krause's papers on localizations contain a solution.:)