Definition 5.1.10 of Olsson's book "Algebraic Spaces and Stacks" gives the definition of an algebraic space as follows:
An algebraic space over a scheme $S$ is a presheaf $X:\text{Sch}/S^{op}\to \text{Set}$ such that
- $X$ is a sheaf in the étale topology
- $\Delta: X \to X\times_S X$ is representable by schemes
- There exists an $S$-scheme $U$ and an étale surjection $U\to X$.
Later, the book states an algebraic space $X$ over $S$ is quasi-separated if the structure morphism $X\to S$ is quasi-separated. However, I don't see any "structure morphism" in the definition of an algebraic space. I understand that it should be a natural transformation $X\to \text{Hom}(-,S)$ but I don't see a natural such map.
Best Answer
Caveat: I don't know anything about algebraic spaces. This answer just seems like the only "natural" interpretation.
$S$ (equipped with the identity map to $S$) is the terminal scheme over $S$. So $\mathrm{Hom}(-,S)$ is the terminal presheaf on $\mathrm{Sch}/S$. Why? For an $S$-scheme $Y$ with structure map $\alpha\colon Y\to S$, $\mathrm{Hom}(Y,S)$ is the singleton set $\{\alpha\}$.
So there is a unique choice of "structure map" $\eta\colon X\to \mathrm{Hom}(-,S)$. Concretely, $\eta_Y\colon X(Y)\to \mathrm{Hom}(Y,S) = \{\alpha\}$ is the constant function with value $\alpha$.