The Stacks project

Definition 78.4.1. Let $B \to S$ as in Section 78.3. Let $U$ be an algebraic space over $B$.

  1. A pre-relation on $U$ over $B$ is any morphism $j : R \to U \times _ B U$ of algebraic spaces over $B$. In this case we set $t = \text{pr}_0 \circ j$ and $s = \text{pr}_1 \circ j$, so that $j = (t, s)$.

  2. A relation on $U$ over $B$ is a monomorphism $j : R \to U \times _ B U$ of algebraic spaces over $B$.

  3. A pre-equivalence relation is a pre-relation $j : R \to U \times _ B U$ such that the image of $j : R(T) \to U(T) \times U(T)$ is an equivalence relation for all schemes $T$ over $B$.

  4. We say a morphism $R \to U \times _ B U$ of algebraic spaces over $B$ is an equivalence relation on $U$ over $B$ if and only if for every $T$ over $B$ the $T$-valued points of $R$ define an equivalence relation on the set of $T$-valued points of $U$.

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