Corollary 5.83. ([Anel et al. 2025, Theorem 2.3.22])
For any acyclic class \(L\), we have
\[D(L) \cap \EffEpi \;=\; \EffEpi \cdot L .\]
In particular, if \(K\) is a congruence we get
\[K^{\epi} = K \cap \EffEpi = \EffEpi \cdot K.\]
Proof
Since \(D(L) = L \cap \Delta^{-1}(L)\), the inclusion “\(\subseteq\)” holds by (2) of the previous lemma. Conversely, Lemma 5.73 gives \(\EffEpi\cdot L\subseteq L\). We also have \(\EffEpi \cdot L = \nabla(L)^m\), so it remains to show that \(\nabla(L) \subseteq \Delta^{-1}(L)\). This is precisely part (1) of the previous lemma.The last statement holds because \(D(K) = K\) for any congruence \(K\).
References
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of $\infty$-topoi III: The acyclic product. 2025.