Remark 77.19.2. A variant of the construction above would have been to sheafify the functor

$\begin{matrix} (\textit{Spaces}/B)^{opp}_{fppf} & \longrightarrow & \textit{Sets}, \\ X & \longmapsto & U(X)/\sim _ X \end{matrix}$

where now $\sim _ X \subset U(X) \times U(X)$ is the equivalence relation generated by the image of $j : R(X) \to U(X) \times U(X)$. Here of course $U(X) = \mathop{\mathrm{Mor}}\nolimits _ B(X, U)$ and $R(X) = \mathop{\mathrm{Mor}}\nolimits _ B(X, R)$. In fact, the result would have been the same, via the identifications of (insert future reference in Topologies of Spaces here).

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).