If \(T\) is a topos, then \(T^{\coh}\) is a local pretopos. In particular, if \(T\) is coherent, then \(T^{\coh}\) is a pretopos.
Proof
Since \(T\) satisfies conditions (1)–(3), it suffices to show that \(T^{\coh}\) is closed under the following operations:
Fiber products: This is clear from the definition of coherent objects.
Finite coproducts: This follows by induction, reducing in degree zero to the stability of quasi-compactness under finite coproducts.
Colimits of groupoids: Let \(X_{\bullet}\) be a groupoid object with realization \(X\). The square is a pullback and \(X_0\to X\) is an effective epimorphism. The descent criterion for relative coherence, applied successively in every degree, therefore shows that \(X\) is coherent whenever \(X_0\) and \(X_1\) are coherent. This is [Lurie 2018, Proposition A.6.1.6].
References
Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.
