The Stacks project

Example 83.4.3. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Here are some standard examples of categories $\mathcal{C}$ that we often come up when applying Definition 83.4.1:

1. $\mathcal{C}$ is the category of all algebraic spaces over $B$,

2. $B$ is separated and $\mathcal{C}$ is the category of all separated algebraic spaces over $B$,

3. $B$ is quasi-separated and $\mathcal{C}$ is the category of all quasi-separated algebraic spaces over $B$,

4. $B$ is locally separated and $\mathcal{C}$ is the category of all locally separated algebraic spaces over $B$,

5. $B$ is decent and $\mathcal{C}$ is the category of all decent algebraic spaces over $B$, and

6. $S = B$ and $\mathcal{C}$ is the category of schemes over $S$.

In this case, if $\phi : U \to X$ is a categorical quotient then we say $U \to X$ is (1) a categorical quotient, (2) a categorical quotient in separated algebraic spaces, (3) a categorical quotient in quasi-separated algebraic spaces, (4) a categorical quotient in locally separated algebraic spaces, (5) a categorical quotient in decent algebraic spaces, (6) a categorical quotient in schemes.