Theorem 5.39. (Anel–Biedermann–Finster–Joyal, [Anel et al. 2022, Proposition 4.2.12])
Let \(T\) be a topos and let \(\Sigma \subseteq \Ar(T)\) be a set of maps. Then the congruence \(\Sigma^c\) is given by
\[\Sigma^c = (\Sigma^{\Delta})^m,\]
the acyclic class generated by the iterated diagonals of \(\Sigma\). This class is of small generation and hence is the left class of a modality.
Proof
Set \(L:=(\Sigma^{\Delta})^m\). Since \(\Sigma^{\Delta}\) is again a set, Proposition 5.32 shows that \(L\) is of small generation. Let \(K:=D^{\infty}(L)\), the largest congruence contained in \(L\).For \(f\in\Sigma\), every iterated diagonal \(\Delta^nf\) belongs to \(\Sigma^{\Delta}\subseteq L\). Hence \(f\in D^n(L)\) for every \(n\), and therefore \(f\in K\). Thus \(K\) is a congruence containing \(\Sigma\), which gives \(\Sigma^c\subseteq K\). Conversely, \(\Sigma^c\) is an acyclic class containing \(\Sigma^{\Delta}\), since it is closed under diagonals by Proposition 5.35. It therefore contains the acyclic closure \(L\). We obtain
\[\Sigma^c\subseteq K\subseteq L\subseteq\Sigma^c,\]
so all three classes agree.References
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. I: Higher sheaves. Adv. Math., 400, 64. 2022.