N coherent objects and topoi

Let \(T\) be a topos. We will inductively define what it means for objects to be \(n\)-coherent for all \(n \geq 0\):

• We say that an object \(X \in T\) is \(0\)-coherent if it is quasi-compact. (This is really a condition on the slice topos \(T_{/X}\).)

• We say that \(T\) is locally \(n\)-coherent if any \(X \in T\) has a cover by \(n\)-coherent objects.

• An object \(X \in T\) is said to be \(n\)-coherent if \(T_{/X}\) is \((n-1)\)-coherent, locally \((n-1)\)-coherent, and for all maps \(Y \to X \leftarrow Z\) such that \(Y\) and \(Z\) are \((n-1)\)-coherent, the pullback \(Y \times_X Z\) is \((n-1)\)-coherent.

• We say \(T\) is \(n\)-coherent if the terminal object is \(n\)-coherent.