nLab
big site

Context

Topos Theory

Could not include topos theory - contents

Big sites

Idea

For CC a site and cCc \in C an object, the over 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.

Definition

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.

Examples

Revised on June 13, 2014 17:00:26 by Anonymous Coward (209.6.246.107)