Example 2

Let Y be a topological space, $(V_i)_{i\in I}$ a family of subsets of Y, and for each $i\in I$, let $(Z_i, p_i)$ be a $V_i$-space. Let X denote the sum topological space of the family $(V_i)_{i\in I}$ and $(Z, p)$ the X-space sum of the family $(Z_i)_{i\in I}$. Let $f: X\rightarrow Y$ be the canonical mapping.

The space $X\times_YX$ is then identified with the sum space of the family $(V_i\cap$ $V_j)_{(i,j)\in I\times I}($I, p. 4, Example 5). Let $\tau$ be a descent datum relative to $f$ on $(Z, p)$. For each pair $(i, j)\in I\times I$, define a continuous mapping $\tau_{i,j}:\overset{-1}{p_{i}}(V_i\cap V_j)\rightarrow \overset{-1}{p_{j}}(V_i\cap V_j)$ by $z\mapsto \tau (z,(p_i(z), j))$. The family $(\tau_{i,j})$ satisfies the following properties:

(i) For each $i\in I$, one has $\tau_{i,i}=$ Id$_{Z_i}$;

(ii) For each pair $(i, j)\in I\times I,\tau_{i,j}$ is an isomorphism of $(V_i\cap V_j)$-spaces ;

(iii) For each triple $(i, j, k)\in I\times I\times I$ and each $z\in \overset{-1}{p_{i}}(V_i\cap V_j\cap V_k)$, one has $\tau_{j,k}(\tau_{i,j}(z)) =\tau_{i,k}(z)$.

Conversely, every family $(\tau_{i,j})$ possessing the above properties arises from a unique descent datum relative to $f$ on $(Z, p)$.