Lemma 6.176.
Let \(T\) be a topos. Then \(T^{\geq \infty}\) is a locus in which every map is an effective epimorphism.
Proof
The subcategory \(T^{\geq \infty}\) of \(T\) spanned by \(\infty\)-connected objects is closed under colimits and finite limits in \(T\) by Proposition 3.39. Since weakly contractible colimits in \(T\) are van Kampen, \(T^{\geq \infty}\) inherits this condition.For any \(X \in T^{\geq \infty}\), the map \(X \to *\) is \(\infty\)-connected by definition. It follows from left cancellation that any map \(X \to Y\) is \(\infty\)-connected, so in particular an effective epimorphism in \(T\). Since \(T^{\geq \infty}\) is closed under finite limits and geometric realizations, it is then also an effective epimorphism in \(T^{\geq \infty}\).