by
S. Awodey

Citations: | 2 - 1 self |

@MISC{Awodey97sheafrepresentation,

author = {S. Awodey},

title = {Sheaf Representation for Topoi},

year = {1997}

}

It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of so-called 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 well-pointed topoi. Completeness theorems for higher-order 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 E, (ii) every stalk of e E is a hyperlocal topos. Moreover, E is boolean just if every stalk of e E is well-pointed. Before defining the term "hyperlocal," we indicate some of the background of the theorem. The original and most familiar sheaf representations are for commutative rings (see [12, ch. 5] for a survey); e.g. a well-known theorem due to Grothendieck [9] asserts that every commutative r...

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