## 81.4 Categorical quotients

This is the most basic kind of quotient one can consider.

Definition 81.4.1. 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.

We often single out a category $\mathcal{C}$ of algebraic spaces over $B$ by some separation axiom, see Example 81.4.3 for some standard cases. Note that $\phi : U \to X$ is a categorical quotient if and only if $U \to X$ is a coequalizer for the morphisms $t, s : R \to U$ in the category. Hence we immediately deduce the following lemma.

Lemma 81.4.2. 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 in algebraic spaces over $B$. If a categorical quotient in the category of algebraic spaces over $B$ exists, then it is unique up to unique isomorphism. Similarly for categorical quotients in full subcategories of $\textit{Spaces}/B$.

Proof. See Categories, Section 4.11. $\square$

Example 81.4.3. 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 81.4.1:

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.

Definition 81.4.4. Let $S$ be a scheme, and let $B$ be an algebraic space over $S$. Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces over $B$ closed under fibre products. Let $j = (t, s) : R \to U \times _ B U$ be pre-relation in $\mathcal{C}$, and let $U \to X$ be an $R$-invariant morphism with $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

1. We say $U \to X$ is a universal categorical quotient in $\mathcal{C}$ if for every morphism $X' \to X$ in $\mathcal{C}$ the morphism $U' = X' \times _ X U \to X'$ is the categorical quotient in $\mathcal{C}$ of the base change $j' : R' \to U'$ of $j$.

2. We say $U \to X$ is a uniform categorical quotient in $\mathcal{C}$ if for every flat morphism $X' \to X$ in $\mathcal{C}$ the morphism $U' = X' \times _ X U \to X'$ is the categorical quotient in $\mathcal{C}$ of the base change $j' : R' \to U'$ of $j$.

Lemma 81.4.5. In the situation of Definition 81.4.1. If $\phi : U \to X$ is a categorical quotient and $U$ is reduced, then $X$ is reduced. The same holds for categorical quotients in a category of spaces $\mathcal{C}$ listed in Example 81.4.3.

Proof. Let $X_{red}$ be the reduction of the algebraic space $X$. Since $U$ is reduced the morphism $\phi : U \to X$ factors through $i : X_{red} \to X$ (insert future reference here). Denote this morphism by $\phi _{red} : U \to X_{red}$. Since $\phi \circ s = \phi \circ t$ we see that also $\phi _{red} \circ s = \phi _{red} \circ t$ (as $i : X_{red} \to X$ is a monomorphism). Hence by the universal property of $\phi$ there exists a morphism $\chi : X \to X_{red}$ such that $\phi _{red} = \phi \circ \chi$. By uniqueness we see that $i \circ \chi = \text{id}_ X$ and $\chi \circ i = \text{id}_{X_{red}}$. Hence $i$ is an isomorphism and $X$ is reduced.

To show that this argument works in a category $\mathcal{C}$ one just needs to show that the reduction of an object of $\mathcal{C}$ is an object of $\mathcal{C}$. We omit the verification that this holds for each of the standard examples. $\square$

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