nLab big site

Idea

For $C$ a site and $c \in C$ an object, the slice category $C/c$ may naturally be thought of as a generalization of the notion of category of open subsets of $c$ in the case of $C =$ Top: its objects are probes of $c$ by arbitrary other objects of $C$.

The over-category naturally inherits the structure of a site itself – this is called the big site of $C$. The corresponding sheaf topos $Sh(C/c)$ is the topos-incarnation of the object $c$.

Definition

Let $C$ be a category equipped with a pretopology $J$ (i.e. a site) and let $a$ be an object of $C$. The slice category $C/a$ inherits a pretopology by setting the covering families to be those collections of morphisms whose image under $C/a \to C$ form a covering family. This is then the big site of $a$.

In the special case that $C$ is some category of spaces with a terminal object $t$, then sheaves on the big site of $t$ form a gros topos. Hence the category of sheaves on the big site of $a$ generalize this idea.

Examples

• For $C$ the category of (affine) schemes, with one of its Grothendieck topologies, notably the etale topology or Zariski topology, for $X$ a scheme, $Sch/X$ is traditionally called the étale site of $X$ or the Zariski site of $X$. See for instance The Stacks project, def. 38.27.3.

Related concepts

• little site