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.