of $S^1$. Section \[sec:IT\] recalls the cluster category $\cC( \cZ )$ from [@IT:cyclicposets sec. 2.4]. Section \[sec:precovering\] provides a main ingredient for the proof of Theorem \[thm:A\] by showing that $\add E( \cX )$ is precovering if and only if $\cX$ satisfies conditions PC1 and PC2. Section \[sec:torsion\_pairs\] proves Theorem \[thm:A\]. Section \[sec:cluster\_tilting\] proves Theorems \[thm:B\] and \[thm:C\].

Admissible subsets of the circle $S^1$ {#sec:admissible_sets}
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\[Cyclically ordered subsets of $S^1$\] \[def:cyclic\] The circle $S^1$, equipped with its usual topology and orientation, has a natural structure as a cyclically ordered set.

We choose anticlockwise as the positive direction, whence the inequalities $x_0 < x_1 < \ldots < x_n$ mean that, when moving anticlockwise around the circle, after encountering $x_{i-1}$ for $i = 1, \ldots, n$, the next element of $\{x_0, \ldots, x_n\}$ encountered is precisely $x_i$. See Figure \[fig:order\]. Soft inequalities are defined analogously.

The cyclic order permits to define closed or (half) open intervals of $S^1$; for instance, the closed interval $[ a,b ]$ is shown in Figure \[fig:order\]. Each interval has an induced linear order.

The cyclic order on $S^1$ induces a cyclic order on each subset of $S^1$, in particular on $\cZ$.

at (60:0.58cm) [$x_0$]{}; (60:0.48) – 