The Stacks project

Definition 83.4.1. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $j = (t, s) : R \to U \times _ B U$ be pre-relation in algebraic spaces over $B$.

1. We say a morphism $\phi : U \to X$ of algebraic spaces over $B$ is a categorical quotient if it is $R$-invariant, and for every $R$-invariant morphism $\psi : U \to Y$ of algebraic spaces over $B$ there exists a unique morphism $\chi : X \to Y$ such that $\psi = \phi \circ \chi $.

2. Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$. Assume $U$, $R$ are objects of $\mathcal{C}$. In this situation we say a morphism $\phi : U \to X$ of algebraic spaces over $B$ is a categorical quotient in $\mathcal{C}$ if $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\phi $ is $R$-invariant, and for every $R$-invariant morphism $\psi : U \to Y$ with $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a unique morphism $\chi : X \to Y$ such that $\psi = \phi \circ \chi $.

3. If $B = S$ and $\mathcal{C}$ is the category of schemes over $S$, then we say $U \to X$ is a categorical quotient in the category of schemes, or simply a categorical quotient in schemes.