The Stacks project

Can we say that a fppf sheaf quotient is a geometric quotient? Here a fppf sheaf quotient means that a morphism of algebraic spaces $U \to X$ is the coequalizer of $s,t : R \rightrightarrows X$ in the category of sheaves on $(Sch/S)_{fppf}$.

I think we can show that a fppf sheaf quotient satisfies (1) and (3). Clearly $U \to X$ $R$-invariant, and we have surjections $R \to U\times_X U$ by \href{https://stacks.math.columbia.edu/tag/046O}{[Stacks,046O]} and $U \to X$ as morphisms of fppf sheaves. This shows that both $R \to U\times_X U$ and $U \to X$ are surjective as algebraic spaces, thus $U \to X$ is an orbit space.

Also $U \to X$ satisfies the condition (4) since from a coequalizer diagram, we obtain an equalizer diagram $$\mathrm{Hom}(X,\mathbb{A}^1) \xrightarrow{} \mathrm{Hom}(U,\mathbb{A}^1) \rightrightarrows \mathrm{Hom}(R,\mathbb{A}^1)$$ and we know that $\mathrm{Hom}(X,\mathbb{A}^1) = \Gamma(X,\mathscr{O}_X)$. This shows that $U \to X$ satisfies the condition (3).

But I'm not sure whether $U \to X$ is submersive. Does it need to be?

@#6489. Yes, I think this is true. Indeed, the condition that $U \to X$ is the quotient of $U$ by $R$ as fppf sheaves is a rather strong one as there is no reason that the fppf sheaf quotient is an algebraic space in general! I agree that using Lemmas 78.19.5 and 83.5.19 it follows that $X$ is an orbit space. (A morphism of algebraic spaces which is surjective as a map of fppf sheaves is also surjective as a morphism of algebraic spaces -- the converse doesn't hold.) I also agree the condition on $R$-invariant functions being functions on $X$ hold, but in order to prove this you have to also do your argument after base change by an etale morphism $X' \to X$ (because you have to check the equality of sheaves on the etale site of $X$). I think submersive works too: namely, since $U \to X$ is surjective as a map of fppf sheaves, there is a surjective flat morphism $a : Y \to X$ of algebraic spaces which is locally of finite presentation such that $a$ lifts to a morphism $b : Y \to U$. Now $Y \to X$ is universally submersive and so therefore is $U \to X$.

But I think the notion of a geometric quotient was defined to deal with cases where it isn't true that the quotient fppf sheaf is representable!

It is clear for me now. Thanks a lot!