Sheaf representation for topoi

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 £, there is a sheaf of categories S on a topological space, such that: (i) £ is equivalent to the category of global sections of £, (ii) every stalk of £ is a hyperlocal topos. Moreover, £ is boolean just if every stalk of £ 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 ring is isomorphic to the ring of global sections of a sheaf of local rings. In Lambek fc Moerdijk [16] it is shown that topoi admit a similar sheaf representation: every topos is equivalent to the topos of global sections of a sheaf of local topoi (cf. also [17, 11.18]. A topos £ is called local if the Heyting algebra Sub^(l) of subobjects of the terminal object 1 of £ has a unique maximal ideal, in analogy with commutative rings. It is easily seen that a topos £ is local iff 1 is indecomposable: for any p,q E Sub£:(l), if p V q = 1 then p = 1 or q = 1. In logical terms, a classifying topos «S[T] for a (possibly * Carnegie Mellon University, Pittsburgh, USA. awodeyOcmu.edu higher-order) theory T is thus local iff the theory T has the "disjunction property": for any T-sentences p,q, if T h p V q then T h p or T h q (cf. §3 below for classifying topoi). A sheaf representation such as those just mentioned yields an embedding theorem, which in the case of topoi yields a logical completeness theorem (just how is shown in §3 below). Prom a logical point of view, however, the local topoi of the Lambek-Moerdijk representation fall short of being those of interest for completeness. For, by other methods, one can already prove logical completeness with respect to a class of topoi that are even more "Setlike" than local ones, in that the terminal object 1 is also projective. Such topoi, in which 1 is both indecomposable and projective, shall here be called hyperlocal In logical terms, a classifying topos S[T\ is hyperlocal iff the theory T has both the disjunction property just mentioned and the so-called existence property: for any type X and any formula T into a topos T that is equivalent to the topos of global sections of a sheaf of hyperlocal topoi. The sheaf representation theorem of this paper thus fits into this pattern of theorems; it states that every topos is equivalent to the topos of global sections of a sheaf of hyperlocal topoi. Moreover, it follows that every boolean topos is equivalent to the topos of global sections of a sheaf of well-pointed topoi. With respect to logical completeness, these are the desired results. The paper is arranged as follows. In §1 it is shown that every topos can be represented as a sheaf of categories on a Grothendieck site (rather than a space). The sheaf in question arises most naturally, not as a sheaf, but as something more general called a "stack." Most of §1 is devoted to the technical problem of turning this (or any) stack into a sheaf. In §2 a recent covering theorem for topoi is used to transport the sheaf constructed in §1 from the site to a space. A comparison of the transported sheaf with the original one then completes the proof of the sheaf representation theorem. In §3 several logical completeness theorems are derived as corollaries. We shall have to do with both small elementary topoi and (necessarily large) Grothendieck topoi. We maintain the convention that "topos" unqualified means the former, but we may still add the qualification "small" for emphasis when called for. We assume familiarity with the basic theory of Grothendieck topoi, e.g. as exposed in [19]. Acknowledgements. The results presented here comprise part of the author's University of Chicago doctoral dissertation, directed by Saunders Mac Lane. Ieke Moerdijk also made essential contributions, and Todd Trimble and Carsten Butz provided helpful advice and comments on an earlier draft of this paper. 1 Slices, stacks, and sheaves Throughout this section, let £ be a fixed small topos. We begin by defining the ^-indexed category £/ (for indexed categories, see [20], [23]). Recall that an £ -indexed category A is essentially the same thing as a pseudofunctor A : £ -» CAT, i.e. a contravariant "functor up to isomorphism" on £ with values in the category CAT of (possibly large) categories. Since the only indexed categories to be considered here are £ -indexed, henceforth indexed category shall mean £ -indexed category. The indexed category £/ : £ -> CAT is defined as follows. For each object / of £, (£I) =df £/I (the slice topos). For each morphism a : J —> I in £, choose a pullback functor