Observation 7.8.
For every topos \(T\), the category \(\Shv_{\CRing}(T)\) admits a factorization system: every map \(A \to B\) factors uniquely as
\[A \to A[S^{-1}] \to B,\]
where the second map is a local map. This factorization is functorial in \(T\).