Example 1

Let X and Y be topological spaces, let $f: X\rightarrow Y$ be a continuous mapping and let $(T, q)$ be a Y-space. Put $Z = X\times_YT$. The mapping $\tau$ from $Z\times_YX$ into Z which associates with $((x, t), x')$ the element $(x', t)$ is a descent datum relative to $f$ on the X-space $(X\times_YT$, pr$_1)$, called the canonical descent datum. For $z_1= (x_1, t_1)$ and $z_2= (x_2, t_2)\in Z$, the relation $R_{\tau}\{z_1, z_2\}$ is equivalent to $t_1=t_2$.