For a site and an object, the over category may naturally be thought of as a generalization of the notion of category of open subsets of in the case of Top: it’s objects are probes of by arbitrary other objects of .
Let be a category equipped with a pretopology (i.e. a site) and let be an object of . The slice category inherits a pretopology by setting the covering families to be those collections of morphisms whose image under form a covering family. This is then the big site of .
In the special case that is some category of spaces with a terminal object , then sheaves on the big site of form a gros topos. Hence the category of sheaves on the big site of generalize this idea.