Exactness in topoi

(1) Writing \(T\) as a left exact localization of a presheaf topos, this follows from the analogous statement in \(\An\), where it is a classical fact. (A model-independent argument for this may be found in [Sattler and Wärn 2025].)(2) Given two diagrams \(X_{\bullet},Y_{\bullet}\colon I \to T\), with \(I\) sifted, we need to show that the canonical map induced by the projections is an isomorphism. We may factor this map as a composite where the first map is induced by the diagonal functor \(I \to I \times I\). Since \(I\) is sifted, this map is final by definition, so the first map is an isomorphism. Moreover, it follows from descent that the functor \(- \times -\colon T \times T \to T\) preserves colimits in both variables separately, so that also the second map is an isomorphism.(3) Given \(A \in \An\), descent gives an equivalence Under this equivalence, the colimit functor \(\colim_A\colon \Fun(A,T) \to T\) corresponds to the forgetful functor \(T_{/\,\colim_A *} \to T\). Limits in a slice are computed in the underlying category whenever the indexing category is nonempty. In particular, this forgetful functor preserves limits indexed by weakly contractible categories, which proves the claim.(4) Let \(P\) denote the constant value of \(\tau_0 Z_{\bullet}\). The comparison map in the statement is a morphism over \(P\). Since \(Z_0 \to P\) is an effective epimorphism, it is enough by Lemma 2.29 to prove that this map becomes an isomorphism after base change along \(Z_0 \to P\).We therefore work in the slice \(T_{/Z_0}\) and suppress this base change from the notation. Each \(Z_n\) is now connected over \(Z_0\), and the degeneracy map \(Z_0 \to Z_n\) supplies a point. Thus \(Z_{\bullet}\) is a simplicial object in the category of pointed connected objects of \(T_{/Z_0}\). Applying the delooping equivalence from Proposition 2.35 degreewise, and using its naturality, we may write for a simplicial group object \(G_{\bullet}\) in \(T_{/Z_0}\). Under the equivalence between objects over \(\bB G_n\) and \(G_n\)-objects from Example 2.13, the simplicial objects \(X_{\bullet}\) and \(Y_{\bullet}\) correspond to simplicial \(G_{\bullet}\)-objects \(X'_{\bullet}\) and \(Y'_{\bullet}\). Degreewise, we then have The quotient on the right is itself a simplicial colimit, namely the realization of the action groupoid. The desired comparison therefore compares the two possible orders of realization of the resulting bisimplicial object. These orders agree by the Fubini theorem for colimits. Moreover, finite products commute with simplicial colimits by part (2), so the products occurring in the action groupoids may also be formed before or after realization. Both sides are consequently identified with the same total colimit, proving the claim after base change and hence in \(T\).