A site is a $\mathcal{U}$-category $\mathcal{C}$ whose objects we call “open sets” and a set $S$ of collections of maps we call “coverings”. A covering of an object $U$ of $\mathcal{C}$ is a small set of morphisms $\left\{{p_{\alpha}:U_{\alpha}\to U}\right\}$ in $\mathcal{C}$. These objects must satisfy the following:

1. 1.

   If $p:U^{{\prime}}\to U$ is an isomorphism, then $\left\{{U^{{\prime}}\overset{p}{\to}U}\right\}$ is a covering.

2. 2.

   If

   $$\left\{{U_{\alpha}\overset{p_{\alpha}}{\to}U}\right\}$$

   is a covering, and for all $\alpha$

   $$\left\{{U_{{\alpha,\beta}}\overset{q_{{\alpha,\beta}}}{\to}U_{\alpha}}\right\}$$

   is also a covering, then

   $$\left\{{U_{{\alpha,\beta}}\overset{p_{\alpha}\circ q_{{\alpha,\beta}}}{\longrightarrow}U}\right\}$$

   is a covering.

3. 3.

   If $\left\{{U_{\alpha}\overset{p_{\alpha}}{\to}U}\right\}$ is a covering, and $V\to U$ is a morphism, then the fibred products $U_{\alpha}\times_{U}V$ exist for all $\alpha$, and we can produce a covering of $V$:

   $$\left\{{V\times_{U}U_{\alpha}\overset{q_{\alpha}}{\to}V}\right\}$$

   where $q_{\alpha}$ is the projection onto the first factor of the fibre product.