nLab
big site

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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: it’s 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 May 8, 2012 10:11:07 by Urs Schreiber (89.204.137.28)