Sheaf representation for topoi (Q1963980)

A result of \textit{J. Lambek} and \textit{I. Moerdijk} [``Two sheaf representations for elementary toposes'', in: The L. E. J. Brower Centen. Symp., Stud. Logic Found. Math. 110, 275-295 (1982; Zbl 0511.03028)] states that every small topos is equivalent to the category of global sections of a sheaf of local topoi. The main interest of this theorem is that it is analogous to a well-known theorem due to \textit{A. Grothendieck} that asserts that every commutative ring is isomorphic to the ring of global sections of a sheaf of local rings. Indeed, a topos is said to be local if the Heyting algebra of subobjects of the terminal object in it has a unique maximal ideal. Another way to express it is to say that a topos is local if the terminal object is indecomposable or, equivalently, if the theory it classifies has the ``disjunction property''. From the logical viewpoint, this corresponds to a completeness theorem of a limited sort. The present paper takes care also of the ``existence property'' by considering topoi, called ``hyperlocal'', in which the terminal object is (not only indecomposable but also) projective and proving a corresponding representation theorem. To prove that every topos is equivalent to the category of global sections of a sheaf of hyperlocal topoi, the sheaf in question arises most naturally as a Giraud stack. Part of the paper is then devoted to the technical question of turning a stack into a sheaf.