properties”) {#sold}
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This section is dedicated to the basics on torsion pairs and $t$-structures. In §\[snotata\] we introduce some notation and recall several important properties of triangulated categories (mostly from [@neebook]).

In §\[shop\] we define and study torsion pairs (in the terminology of [@aiya]). Our results are rather easy; yet the author does not know any references for most of them. In §\[sts\] we recall some basics on $t$-structures and relate them to torsion pairs. We also study the notions of $t$-projective objects for the purpose of using them in §\[sadjw\] and later.

Some categorical preliminaries {#snotata}
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When we will write $i\ge c$ or $i\le c$ (for some $c\in {{\mathbb{Z}}}$) we will mean that $i$ is an integer satisfying this inequality.

A cardinal ${\alpha}$ is said to be [*regular*]{} if it cannot be presented as a sum of less then ${\alpha}$ cardinals that are less than ${\alpha}$.

Most of the categories of this paper will be locally small. When considering a category that is not locally small we will usually say that it is (possibly) big; we will not need much of these categories. For categories $C,D$ we write $D\subset C$ if $D$ is a full subcategory of 