Remark 5.65.
In [Anel et al. 2025, Remark 2.1.34], the authors make the following analogy with ring theory. Given a morphism \(\varphi\colon A\to B\) of commutative rings with kernel \(I\), its usual image factorization is
\[A\longrightarrow A/I\lhook\joinrel\longrightarrow B.\]
The injectivity of the second map is analogous to the conservativity of \(\varphi^{\cons}\colon T/K\to S\). Let \(W\subseteq A\) be the multiplicative subset of elements whose images in \(A/I\) are units. Then the quotient map \(A\to A/I\) factors further as
\[A\longrightarrow A[W^{-1}]\longrightarrow A/I.\]
Altogether, this gives the triple factorization
\[A\longrightarrow A[W^{-1}]\longrightarrow A/I\lhook\joinrel\longrightarrow B,\]
which the authors compare with the factorization of \(T\to S\) through \(T/K^{\mono}\) and \(T/K\).
References
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of $\infty$-topoi III: The acyclic product. 2025.