Lemma 4.44. ([Uemura 2025, Lemma 4.3])
The wide subcategory \(\Fam(T) \subseteq \Ar(T)\) is closed under small colimits and pullbacks.
Proof
The assertion about colimits is precisely the topos direction of Proposition 2.16, after identifying \(\Fam(T)\) with \(\Ar^{\pb}(T)\).For pullbacks, note that \(\Fam(T)\) admits pullbacks if and only if for each \(u \in \Fam(T)\) the slice category \(\Fam(T)_{/u}\) admits binary products, which is immediate from Lemma 4.43.
References
- Taichi Uemura. Colimits in the ∞-category of ∞-topoi and étale morphisms. 2025.