formula $\{\vec{x}, \vec{x_{i}}. \theta'\}$ from $\{\vec{x}. \phi\}$ to $\{\vec{x_{i}}. \phi_{i}\}$ such that $\phi \vdash_{\vec{x}} (\exists \vec{x_{i}})(\theta' \wedge \theta_{i})$ is provable in $\mathbb T$;

\(iv) we say that $\phi(\vec{x})$ is $\mathbb T$-compact if for any family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of geometric formulae in the same context, $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ provable in $\mathbb T$ implies $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I'}} \psi_{i}$ provable in $\mathbb T$ for some finite subset $I'$ of $I$;

\(v) we say that $\phi(\vec{x})$ is $\mathbb T$-supercompact if for any family $\{\psi_{i}(\vec{x}) \textrm{ | } i\in I\}$ of geometric formulae in the same context, $\phi \vdash_{\vec{x}} \mathbin{\mathop{\textrm{\huge $\vee$}}\limits_{i\in I}} \psi_{i}$ provable in $\mathbb T$ implies $\phi \vdash_{\vec{x}} \psi_{i}$ provable in $\mathbb T$ for some $i\in I$.

\[Onetoposmany\_equiv\] Let $\mathbb T$ be a geometric theory over a signature $\Sigma$ and $\phi(\vec{x})$ a geometric formula-in-context over $\Sigma$. Then

\(i) $\phi(\vec{x})$ is $\mathbb T$-complete if and only if $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ is an atom of\
${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$;

\(ii) $\phi(\vec{x})$ is $\mathbb T$-indecomposable if and only if $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ is an indecomposable object of ${{\bf Sh}}({\cal C}_{\mathbb T}^{\textrm{geom}}, J^{\textrm{geom}}_{\mathbb T})$;

\(iii) $\phi(\vec{x})$ is $\mathbb T$-irreducible if and only if $y^{\textrm{geom}}_{\mathbb T}(\{\vec{x}. \phi\})$ 