The Stacks project

Definition 83.5.4. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation over $B$. Let $\mathop{\mathrm{Spec}}(k) \to B$ be a geometric point of $B$.

  1. We say $\overline{u}, \overline{u}' \in U(k)$ are weakly $R$-equivalent if they are in the same equivalence class for the equivalence relation generated by the relation $j(R(k)) \subset U(k) \times U(k)$.

  2. We say $\overline{u}, \overline{u}' \in U(k)$ are $R$-equivalent if for some overfield $k \subset \Omega $ the images in $U(\Omega )$ are weakly $R$-equivalent.

  3. The weak orbit, or more precisely the weak $R$-orbit of $\overline{u} \in U(k)$ is set of all elements of $U(k)$ which are weakly $R$-equivalent to $\overline{u}$.

  4. The orbit, or more precisely the $R$-orbit of $\overline{u} \in U(k)$ is set of all elements of $U(k)$ which are $R$-equivalent to $\overline{u}$.

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