Definition 6.81. (Relative shape)
Let \(\phi_*\colon T \to S\) be a morphism of topoi. We define the relative shape of \(\phi_*\) as the pro-object
\[\abs{\phi} \quad:=\quad \phi_\sharp(*) \qquad\in \Pro(S).\]
Note that this pro-object corepresents the composite functor \(\Gamma_{T} \circ \phi^*\colon S \to \An\), where \(\Gamma_{T}\colon T \to \An\) is the global sections functor of \(T\).
We say that \(\phi_*\) is of constant shape if \(\abs{\phi}\) is pro-constant, i.e., if it is corepresentable by an object of \(S\).