Morphisms representable by algebraic spaces

Let $F\to G$ be a morphism of presheaves of sets on $(Sch\mathrm{/}S{)}_{fppf}$ (Fppf sheaves of sets and sheafification, Natural transformation and its components). It is representable by algebraic spaces when for every $S$-scheme $T$ and every morphism $T\to G$ the fibre product $F{\times }_{G}T$ is an algebraic space over $S$ (Algebraic spaces over a scheme, defined as fppf sheaves, Fibre product of schemes). This generalizes representability by schemes (Representable morphisms of presheaves and fibrewise properties), under the Axiom of Choice (The Axiom of Choice) of Every representable functor is an algebraic space: every morphism representable by schemes is representable by algebraic spaces, because a scheme is an algebraic space, and the fibre products agree. A property of morphisms of algebraic spaces that is stable under base change (Morphisms, products and fibre products of algebraic spaces) is attributed fibrewise to such a morphism: it has property $\mathcal{P}$ when every base change $F{\times }_{G}T\to T$ is a morphism of algebraic spaces with $\mathcal{P}$. In this way one speaks of representable etale, smooth, flat, surjective, open-immersion and closed-immersion morphisms of presheaves.

A 1-morphism of stacks in groupoids $f\text{\hspace{0.17em}⁣}:\mathcal{X}\to \mathcal{Y}$ over $(Sch\mathrm{/}S{)}_{fppf}$ (Descent data, prestacks and stacks in groupoids over the fppf site) is representable by algebraic spaces when for every $S$-scheme $T$ and every 1-morphism $T\to \mathcal{Y}$ — equivalently, by the 2-Yoneda lemma, for every object $x\in {\mathcal{Y}}_T$ — the 2-fibre product $\mathcal{X}{\times }_{\mathcal{Y},T}$ is equivalent to the stack in setoids ${\mathcal{S}}_Z$ of an algebraic space $Z$ over $T$. Smooth, etale, surjective and other fibrewise properties are then defined by base change to schemes, so the definition specializes to the presheaf case when $\mathcal{X},\mathcal{Y}$ are stacks in setoids of presheaves of sets.