Sheaf relative to a presheaf

Question

I was looking at these notes https://arxiv.org/pdf/math/0503247.pdf and it keeps using the phrase "sheaf relative to a presheaf" (i.e. definition 3.3). What does this mean? An online search has not revealed anything and I can't make any sense out of this.

Answer 1

Basically, it means nothing. The author wants to highlight with "When $X$ is relative to $S$" that a certain construction $F(X,S)$ (depending on these objects) results in an object which is defined over $S$, i.e. admits a morphism to $S$. (This is the usual meaning in algebraic geometry of being "over $S$" or "relative to $S$".)

In the case of the classifying stack of a group $G$ (which is Definition 3.3 in the mentioned paper) we can take any base scheme $S$ and define $[S/G]$ as the stack associated to the "trivial" groupoid $[G \times S \rightrightarrows S]$. And even more generally, if $S$ is any presheaf (on the given site $\mathsf{C}$ in the paper) of sets and $G$ is a presheaf of groups, we can define $[S/G]$ as the sheaf associated to $[G \times S \rightrightarrows S]$, which thus is a sheaf of sets (on the same site) and clearly admits a morphism to $S$. Notice that we have $[S/G] = S \times [\star/G] = S \times BG$.