Proposition 6.113.
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If \((C,\tau)\) is a finitary Grothendieck site and \(C\) admits fiber products, then \(\Shv_{\tau}(C)\) is locally coherent. If \(C\) also admits a terminal object, then \(\Shv_{\tau}(C)\) is also coherent.
If \(T\) is hypercomplete and locally coherent, then there exists a finitary site \((C,\tau)\) with fiber products such that \(T \simeq \Shv_{\tau}(C)^{\hyp}\). If \(T\) is also coherent, then we can choose \(C\) to have a terminal object.
Proof
For (1), let \(a\colon C\to\Shv_\tau(C)\) denote sheafified Yoneda. Every sheaf is covered by objects in the image of \(a\). Finitariness says precisely that each \(a(c)\) is quasi-compact: a cover of \(a(c)\) corresponds to a covering sieve on \(c\), which admits a finite covering refinement. Since \(C\) admits fiber products and \(a\) preserves them, the image of \(a\) is closed under fiber products. The claim now follows from Corollary 6.106. If \(C\) has a terminal object, its image is terminal, so the resulting topos is coherent.For (2), choose a small full subcategory \(C\subseteq T^{\coh}\) which covers \(T\) and is closed under fiber products; in the coherent case we also arrange that \(C\) contains the terminal object, using Remark 6.107. Give \(C\) the topology in which a family \((U_i\to U)\) covers if \(\bigsqcup_iU_i\to U\) is an effective epimorphism in \(T\). Since \(U\) is quasi-compact, every covering family admits a finite subfamily which still covers, so this topology is finitary.Let \(u\colon C \hookrightarrow T\) denote the inclusion. Since \(C\) is closed under fiber products, Proposition 2.43 shows that the left Kan extension
\[u_!\colon \PSh(C) \to T\]
is left exact. By definition of \(\tau\), all generating \(\tau\)-covering sieves are sent to effective epimorphisms in \(T\), so \(u_!\) factors as \[\PSh(C) \to \Shv_{\tau}(C) \to T.\]
On the other hand, Lemma 6.57 shows that the restriction functor \(T \to \PSh(C)\) is fully faithful. Since its essential image consists of \(\tau\)-sheaves, the induced functor \(T \to \Shv_{\tau}(C)\) is also fully faithful. Thus \(\Shv_{\tau}(C) \to T\) is a left exact localization. It is epigenic because its kernel is generated by maps from covering sieves to representables, all of which become effective epimorphisms in \(T\). An epigenic quotient induces an equivalence on hypercompletions by [Anel et al. 2025, Lemma 2.1.33]. Since \(T\) is hypercomplete, we conclude that \(T \simeq \Shv_{\tau}(C)^{\hyp}\).References
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of $\infty$-topoi III: The acyclic product. 2025.