Definition 5.66.
Given a congruence \(K\) of small generation, we define its hypercompletion as the kernel of the composite:
\[K^{\hyp} \quad := \quad \ker \bigl( T \to T/K \to (T/K)^{\hyp} \bigr),\]
where \((-)^{\hyp}\) denotes the hypercompletion of a topos (localization at \(\infty\)-connected maps). Note that we always have \(K \subseteq K^{\hyp}\). We say a congruence \(K\) is hypercomplete if \(K = K^{\hyp}\).