Summary: Sheaf Representation for Topoi
S. Awodey \Lambda
It is shown that every (small) topos is equivalent to the category of
global sections of a sheaf of socalled hyperlocal topoi, improving on
a result of Lambek & Moerdijk. It follows that every boolean topos
is equivalent to the global sections of a sheaf of wellpointed topoi.
Completeness theorems for higherorder logic result as corollaries.
The main result of this paper is the following.
Theorem (Sheaf representation for topoi). For any small topos E,
there is a sheaf of categories e
E on a topological space, such that:
(i) E is equivalent to the category of global sections of e
(ii) every stalk of e
E is a hyperlocal topos.
Moreover, E is boolean just if every stalk of e
E is wellpointed.
Before defining the term ``hyperlocal,'' we indicate some of the back