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Big sites


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

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


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

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


Last revised on November 15, 2022 at 22:03:18. See the history of this page for a list of all contributions to it.