ScalingStacks

Example B.1.8.

Let \(\mathbb N^\times \coloneqq \{1, 2, 3, \ldots \}^\times\) denote the (commutative) monoid of natural numbers under multiplication. Given two elements \(s,t \in \mathbb N^\times\), their corresponding morphisms in \(B \mathbb N^\times\) satisfy \(s \bot t\) (and thereafter \(t \bot s\)) if and only if \(s\) and \(t\) are coprime. From here, it is easy to check that e.g. the pairs (powers of 2, odds) and (odds, powers of 2) define factorization systems on \(B \mathbb N^\times\). More generally, if \(\{2, 3, 5, \ldots \} = P_1 \sqcup P_2\) denotes a two-element partition of the set of prime numbers, then \[\text{(powers of elements of $P_1$, powers of elements of $P_2$)}\] determines a factorization system on \(B \mathbb N^\times\), and moreover every factorization system on \(B \mathbb N^\times\) arises in this way.