Lemma 93.14.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Then $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is an algebraic stack, and is a product in the $2$-category of algebraic stacks over $S$.

## 93.14 2-Fibre products of algebraic stacks

The $2$-category of algebraic stacks has products and $2$-fibre products. The first lemma is really a special case of Lemma 93.14.3 but its proof is slightly easier.

**Proof.**
An object of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ over $T$ is just a pair $(x, y)$ where $x$ is an object of $\mathcal{X}_ T$ and $y$ is an object of $\mathcal{Y}_ T$. Hence it is immediate from the definitions that $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is a stack in groupoids. If $(x, y)$ and $(x', y')$ are two objects of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ over $T$, then

Hence it follows from the equivalences in Lemma 93.10.11 and the fact that the category of algebraic spaces has products that the diagonal of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is representable by algebraic spaces. Finally, suppose that $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and let $x, y$ be surjective smooth morphisms $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$, $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. Note that

The object $(\text{pr}_ U^*x, \text{pr}_ V^*y)$ of $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ over $(\mathit{Sch}/U \times _ S V)_{fppf}$ thus defines a $1$-morphism

which is the composition of base changes of $x$ and $y$, hence is surjective and smooth, see Lemmas 93.10.6 and 93.10.5. We conclude that $\mathcal{X} \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{Y}$ is indeed an algebraic stack. We omit the verification that it really is a product. $\square$

Lemma 93.14.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{Z}$ be a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ whose diagonal is representable by algebraic spaces. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of stacks in groupoids. Then the $2$-fibre product $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack.

**Proof.**
We have to check conditions (1), (2), and (3) of Definition 93.12.1. The first condition follows from Stacks, Lemma 8.5.6.

The second condition we have to check is that the $\mathit{Isom}$-sheaves are representable by algebraic spaces. To do this, suppose that $T$ is a scheme over $S$, and $u, v$ are objects of $(\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y})_ T$. By our construction of $2$-fibre products (which goes all the way back to Categories, Lemma 4.32.3) we may write $u = (x, y, \alpha )$ and $v = (x', y', \alpha ')$. Here $\alpha : f(x) \to g(y)$ and similarly for $\alpha '$. Then it is clear that

is a cartesian diagram of sheaves on $(\mathit{Sch}/T)_{fppf}$. Since by assumption the sheaves $\mathit{Isom}(y, y')$, $\mathit{Isom}(x, x')$, $\mathit{Isom}(f(x), g(y'))$ are algebraic spaces (see Lemma 93.10.11) we see that $\mathit{Isom}(u, v)$ is an algebraic space.

Let $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and let $x, y$ be surjective smooth morphisms $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$, $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. Consider the morphism

As the diagonal of $\mathcal{Z}$ is representable by algebraic spaces the source of this arrow is representable by an algebraic space $F$, see Lemma 93.10.11. Moreover, the morphism is the composition of base changes of $x$ and $y$, hence surjective and smooth, see Lemmas 93.10.6 and 93.10.5. Choosing a scheme $W$ and a surjective étale morphism $W \to F$ we see that the composition of the displayed $1$-morphism with the corresponding $1$-morphism

is surjective and smooth which proves the last condition. $\square$

Lemma 93.14.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of algebraic stacks. Then the $2$-fibre product $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack. It is also the $2$-fibre product in the $2$-category of algebraic stacks over $(\mathit{Sch}/S)_{fppf}$.

**Proof.**
The fact that $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack follows from the stronger Lemma 93.14.2. The fact that $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is a $2$-fibre product in the $2$-category of algebraic stacks over $S$ follows formally from the fact that the $2$-category of algebraic stacks over $S$ is a full sub $2$-category of the $2$-category of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$.
$\square$

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