compactly generated if such a generating set ${\mathcal{P}}$ exists).[^8] 1. Then the heart ${{\underline{Ht}}}$ of $t$ has an injective cogenerator and satisfies the AB3\* axiom.

2\. Assume in addition either that ${\underline{C}}$ is the homotopy category of a proper simplicial stable model category or that $t$ is non-degenerate. Then ${{\underline{Ht}}}$ is a Grothendieck abelian category.

Past 2 of this theorem answers Question 3.8 of [@parrasao] “in all reasonable cases”;[^9] part 1 appears to be new also.

Now we describe a result that was important for the proof of part 1 of Theorem \[tgroth\].

One of the main topics of this paper is the study of the relation of (smashing) weight structures to perfect sets of objects (that we will now define; recall that these are also closely related to the Brown representability condition).[^10]

\[textw\] Assume that ${\underline{C}}$ is closed with respect to small coproducts. Let ${\mathcal{P}}$ be a [*perfect*]{} set of objects of ${\underline{C}}$ (i.e., assume that the class [*${\mathcal{P}}$-null*]{} of those morphisms that are annihilated by corepresentable functors of the type ${\underline{C}}(P,-)$ for $P\in {\mathcal{P}}$ is closed with respect to coproducts).[^11] Then the couple $w=(L,R)$ is a smashing weight structure, where $R=\cap_{i<0}({\mathcal{P}}{{}^{\perp}}[i])$ and $L=({{}^{\perp}}R)[1]$. Moreover, the class $L$ may 