Definition 1

A descent data relative to $f$ on the X-space $(Z, p)$ is called a continuous mapping $\tau : Z\times_YX\rightarrow Z$ satisfying the following two properties:

(i) For every pair $(x, x')\in X\times_YX$, the mapping $z\mapsto \tau (z, x')$ induces by restriction a bijection $\tau_{x,x'}$ of $Z_x$ onto $Z_{x'}$;

(ii) For every triplet $(x, x', x'')$ of points of X such that $f(x) =$ $f(x') =f(x'')$, one has

If $\tau$ is a descent data on $(Z, p)$, the family $(\tau_{x,x'})$, for $(x, x')\in X\times_YX$, is thus a law of operation (on the right) of the groupoid $X\times_YX$ on the X-space $(Z, p)$ (II, p. 167). In particular, one has $\tau_{x,x}=$ Id$_{Z_x}$ for every $x\in X$, which is also written $\tau (z, p(z)) =z$ for every $z\in Z$. Conversely, given a law of operation (on the right) of the groupoid $X\times_YX$ on $(Z, p)$, the map from $Z\times_YX$ into Z defined by $(z, x)\mapsto z\cdot (p(z), x)$ verifies the relations (i) and (ii) of the definition.

Let X and Y be topological spaces, $f: X\rightarrow Y$ a continuous mapping and let $(Z, p)$ be an X-space. If $\tau$ is a descent data relative to $f$ on $(Z, p)$, the relation $R_{\tau}(z_1, z_2)$ defined by « $f(p(z_1)) =$ $f(p(z_2))$ and $\tau (z_1, p(z_2)) =z_2$ » is the equivalence relation in Z deduced from the operation of the groupoid $X\times_YX$ defined by $\tau$; it is compatible with the mapping $f\circ p$. It is said that this is the equivalence relation associated with the descent data $\tau$. The canonical bijection $(z_1, z_2)\mapsto (z_1, p(z_2))$ of the graph Γ of the equivalence relation $R_{\tau}$ onto $Z\times_YX$ is a homeomorphism, the reciprocal homeomorphism applies an element $(z_1, x_2)\in Z\times_YX$ to $(z_1, \tau (z_1, x_2))$.

Conversely, let R be an equivalence relation in Z compatible with the mapping $f\circ p: Z\rightarrow Y$ and let Γ be the graph of R. Suppose moreover that the mapping $p_2: (z_1, z_2)\mapsto (z_1, p(z_2))$ defines a homeo-morphism of Γ onto $Z\times_YX$. The mapping $\tau : Z\times_YX\rightarrow Z$ given by pr$_2\circ p^{-1}_2$ is continuous; it is a descent data relative to $f$ on $(Z, p)$ and the relation R is the equivalence relation associated with $\tau$.