Algebraic spaces over a scheme, defined as fppf sheaves

An algebraic space over $S$ is a presheaf of sets $F$ on $(Sch\mathrm{/}S{)}_{fppf}$ (Fppf sheaves of sets and sheafification) such that:

1. $F$ is an fppf sheaf;
2. the diagonal morphism $F\to F\times F$ is representable by schemes (Representable morphisms of presheaves and fibrewise properties);
3. there exists an $S$-scheme $U$ (Schemes) together with a morphism $h_U\to F$ from the presheaf $h_U={\mathrm{Mor}}_S(-,U)$ represented by $U$ (Presheaves, covariantly and contravariantly representable functors, and representations) which is representable, etale and surjective (Étale morphism of schemes, Morphisms of schemes).

A morphism of algebraic spaces over $S$ is a natural transformation of the underlying presheaves; algebraic spaces over $S$ form a full subcategory of the presheaves of sets on the fppf site. Under the Axiom of Choice (The Axiom of Choice) inherited from represented-sheaf descent, a scheme $X$ over $S$ gives an algebraic space $h_X$, and $X\mapsto h_X$ is a full embedding (Every representable functor is an algebraic space ↗). No separatedness, quasi-compactness, finiteness or Noetherian hypothesis is part of the definition: condition 3 asks only for a single etale scheme cover, not for a Zariski cover or for quasi-compactness, and the covering morphism may have infinite index set.

An etale scheme cover of an algebraic space $F$ is a morphism $h_U\to F$ as in condition 3; its existence is part of the definition, while a second such cover is compared with the first by the fibrewise properties of representable morphisms of Representable morphisms of presheaves and fibrewise properties.