Proposition 6.86. (Aizenbud and Carmeli (2021, Proposition 3.1.5))
Let \(\phi_* \colon T \to S\) be a morphism of topoi and let \(\{U_i\}_{i \in I}\) be a collection of objects which generates \(T\) under colimits. Then the following conditions are equivalent:
The morphism \(\phi_*\) is essential;
Every object \(U_i\) has constant shape relative to \(\phi_*\).
Proof
Recall from Example 6.78 that the étale morphism \(j_*\colon T_{/U} \to T\) is essential. Pro-left adjoints compose, so the relative shape of the composite \(\phi_* \circ j_*\colon T_{/U} \to S\) is the image of the terminal object of \(T_{/U}\) under the composite
\[T_{/U} \xrightarrow{j_\sharp} T \xrightarrow{\phi_\sharp} \Pro(S).\]
Since \(j_\sharp\) is the forgetful functor sending \((V \to U)\) to \(V\), we see that the relative shape of \(\phi_* \circ j_*\) is equivalent to \(\phi_\sharp(U) \in \Pro(S)\).If \(\phi_*\) is essential, then \(\phi_\sharp\) factors through \(S\), so \(\phi_\sharp(U)\) is pro-constant for all \(U \in T\), and in particular for all \(U_i\).Conversely, assume that \(\phi_\sharp(U_i)\) is pro-constant for all \(i \in I\). The embedding \(S\hookrightarrow\Pro(S)\) and the functor \(\phi_\sharp\colon T\to\Pro(S)\) both preserve colimits. Since the collection \(\{U_i\}\) generates \(T\) under colimits, it follows that \(\phi_\sharp\) factors through \(S\). By Proposition 6.80, this means that \(\phi_*\) is essential.References
- Avraham Aizenbud, Shachar Carmeli. Relative de Rham theory on Nash manifolds. arXiv preprint arXiv:2105.12417. 2021.