Remark 5.40.
The same formula holds for an arbitrary class \(\Sigma\), see [Anel et al. 2022, Proposition 4.2.12]. One writes \(\Sigma\) as the union of a filtered system of sets closed under taking the diagonals that have already appeared and then uses that both acyclic and congruence closure preserve these filtered unions. The set case above is the form needed in these notes, and it has the additional advantage that \((\Sigma^{\Delta})^m\) automatically defines a modality.
References
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of ∞-topoi. I: Higher sheaves. Adv. Math., 400, 64. 2022.