of [@neebook]. The second part is immediate from Theorem 8.4.4 of ibid. (combined with Proposition \[pbouloc\](I)).

On torsion pairs {#shop}
----------------

As we have already said, this paper is mostly dedicated to the study of weight structures and $t$-structures. Now, these notions have much in common; so we start from recalling an (essentially) more general definition of a torsion pair (in the terminology of [@aiya Definition 1.4]; in [@postov Definition 3.2] torsion pairs were called complete Hom-orthogonal pairs).[^17]

\[dhop\]

A couple $s$ of classes ${\mathcal{LO}},{\mathcal{RO}}\subset{\operatorname{Obj}}{\underline{C}}$ (of $s$-left orthogonal and $s$-right orthogonal objects, respectively) will be said to be a [*torsion pair*]{} (for ${\underline{C}}$) if ${\mathcal{LO}}^{\perp}={\mathcal{RO}}$, ${\mathcal{LO}}={}^{\perp}{\mathcal{RO}}$, and for any $M\in{\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $$\label{swd}
L_sM\stackrel{a_M}{\to} M\stackrel{n_M}{\to} R_sM{\to} L_sM[1]$$ such that $L_sM\in {\mathcal{LO}}$ and $ R_sM\in {\mathcal{RO}}$. We will call any triangle of this form an [*$s$-decomposition*]{} of $M$; $a_M$ will be called an [*$s$-decomposition morphism*]{}.

We will also need the following auxiliary definitions.

\[dhopo\] Let $s=({\mathcal{LO}},{\mathcal{RO}})$ be a torsion pair.

1\. We will say that $s$ is [*coproductive*]{} (resp. [*productive*]{}) if ${\mathcal{RO}}$ is coproductive (resp. ${\mathcal{LO}}$ is productive). We will also say that $s$ is smashing (resp. cosmashing) if ${\underline{C}}$ in addition has coproducts (resp. products).

We will 