Let $X$ be a scheme. For $Y$ a scheme over $X$, the representable presheaf $h\_Y : U\mapsto \mathrm{Hom}\_X(U,Y)$ on the small étale site $X\_{et}$ is actually a sheaf, and by the Yoneda lemma the mapping $Y\to h\_Y$ is fully faithful when restricted to the category of schemes étale over $X$. Is there a bigger subcategory of the schemes over $X$ such that it is still fully faithful ?


Similarly, what about algebraic spaces over $X$ ? Seing those as sheaves on the big fppf site of $X$, we can restrict them to the small étale site. This is operation is fully faithful when done on algebraic spaces that are étale schemes, can we get a bigger subcategory for which it is still fully faithful ?



Extending the domain of the yoneda embedding map from étale schemes to the small étale topos so that it is still fully faithful
The good setting to answer this question really is that of algebraic spaces. Then I believe that the answer is no, basically because every sheaf $F$ on the small étale site is representable by an étale $X$-algebraic space. The algebraic space that represents $F$ is constructed as follows: consider the collection of pairs $(V,x)$ with $V$ an affine étale $X$-scheme and $x\in F(V)$. Let $U$ be the disjoint sum of the $V$'s indexed by this collection. If $(V,x)$ and $(V',x')$ are two pairs, the natural morphism $V\times\_FV'\to V\times\_XV'$ is an open immersion. This shows that $U\times\_FU$ is open in $ U\times\_X U$ and hence defines an étale equivalence relation on $U$. The $X$-algebraic space we're looking for is the quotient of this equivalence relation.


This is due to Michael Artin, reference: his book "Théorèmes de représentabilité pour les espaces algébriques", chap. VII. Le théorème de finitude en cohomologie étale, §1. Définition des faisceaux constructibles.


