Giraud characterizations of topoi

(1) \(\implies\) (2): Clear, since \((-)^{\simeq}\colon \Cat \to \An\) preserves limits.(2) \(\implies\) (1): Assume (2), and let \(X_{\bullet}\colon I \to T\) be a diagram with colimit \(X\). By universality of colimits, the comparison functor is fully faithful by Lemma 2.8. To prove essential surjectivity, let \(U\) be an object classifier and choose an object of the limit on the right. Passing to cores and using the classifying property of \(U\), this object determines a point of The final isomorphism follows because \(U\colon T\catop \to \widehat{\An}\) preserves limits. The resulting map \(X \to U\) classifies an object over \(X\) whose pullback to every \(X_i\) is isomorphic to the prescribed object. Thus the comparison functor is essentially surjective and hence an equivalence.(1) \(\implies\) (3): We verify the three conditions:

• Universality of colimits holds by definition of a topos.
• Coproducts are disjoint by Example 2.12.
• All groupoids are effective by Corollary 2.23.

(4) \(\implies\) (1): Since \(\An\) is a topos, it follows that \(\PSh(C) = \Fun(C\catop,\An)\) is a topos, and hence so is any left exact localization of \(\PSh(C)\) by Corollary 2.17.(3) \(\implies\) (4): We use the proposition below. Choose a sufficiently large regular cardinal \(\kappa\) such that \(T\) is generated under colimits by a small full subcategory \(C\) of \(\kappa\)-compact objects and such that \(C\) is closed under finite limits. Such a choice is possible because finite limits in a presentable category are accessible. The inclusion \(C \hookrightarrow T\) is then left exact, so the proposition shows that its colimit-preserving extension is left exact. Its right adjoint is the restricted Yoneda functor \(X\mapsto\Hom_T(-,X)|_C\), which is fully faithful because \(C\) generates \(T\) under colimits. Thus \(F\) is a left exact localization, giving (4).