The Stacks project

Proof. It is clear that (2) implies (3). Parts (1) and (2) are equivalent by Lemma 94.12.4. Hence it suffices to prove (2). First, we note that $\mathcal{S}_ F$ is stack in sets since $F$ is a sheaf (Stacks, Lemma 8.6.3). A fortiori it is a stack in groupoids. Second the diagonal morphism $\mathcal{S}_ F \to \mathcal{S}_ F \times \mathcal{S}_ F$ is the same as the morphism $\mathcal{S}_ F \to \mathcal{S}_{F \times F}$ which comes from the diagonal of $F$. Hence this is representable by algebraic spaces according to Lemma 94.9.4. Actually it is even representable (by schemes), as the diagonal of an algebraic space is representable, but we do not need this. Let $U$ be a scheme and let $h_ U \to F$ be a surjective étale morphism. We may think of this as a surjective étale morphism of algebraic spaces. Hence by Lemma 94.10.3 the corresponding $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ is surjective and étale. $\square$

Proof. The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 94.13.1. Finally, assume (1). By Stacks, Lemma 8.6.3 there exists a sheaf $F$ on $(\mathit{Sch}/S)_{fppf}$ and an equivalence $j : \mathcal{X} \to \mathcal{S}_ F$. By Lemma 94.9.5 the fact that $\Delta _\mathcal {X}$ is representable by algebraic spaces, means that $\Delta _ F : F \to F \times F$ is representable by algebraic spaces. Let $U$ be a scheme, and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a surjective étale morphism. The composition $j \circ x : (\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ corresponds to a morphism $h_ U \to F$ of sheaves. By Bootstrap, Lemma 80.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 94.10.4 we conclude that $h_ U \to F$ is surjective and étale. Finally, we apply Bootstrap, Theorem 80.6.1 to see that $F$ is an algebraic space. $\square$

Proof. The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 94.13.2. Finally, assume (1). By Stacks, Lemma 8.6.3 there exists an equivalence $j : \mathcal{X} \to \mathcal{S}_ F$ where $F$ is a sheaf on $(\mathit{Sch}/S)_{fppf}$. By Lemma 94.9.5 the fact that $\Delta _\mathcal {X}$ is representable by algebraic spaces, means that $\Delta _ F : F \to F \times F$ is representable by algebraic spaces. Let $U$ be a scheme and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a surjective smooth morphism. The composition $j \circ x : (\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ corresponds to a morphism $h_ U \to F$ of sheaves. By Bootstrap, Lemma 80.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 94.10.4 we conclude that $h_ U \to F$ is surjective and smooth. In particular it is surjective, flat and locally of finite presentation (by Lemma 94.10.9 and the fact that a smooth morphism of algebraic spaces is flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 67.37.5 and 67.37.7). Finally, we apply Bootstrap, Theorem 80.10.1 to see that $F$ is an algebraic space. $\square$