ov 2 00 7 Sheaves as modules ∗

We revisit sheaves on locales by placing them in the context of the theory of Hilbert quantale modules. The local homeomorphisms p : X → B are identified with the locales X that are Hilbert B-modules equipped with a natural notion of basis. These modules form a full subcategory B-HMB of the category of Hilbert B-modules where all the homomorphisms are adjointable: for each homomorphism h there is a unique adjoint h defined by 〈h(x), y〉 = 〈x, h(y)〉, and this makes B-HMB a self-dual category. The local homeomorphism associated to a sheaf is obtained in a point-free way by completing the set of local sections with respect to the restriction order. The equivalence of the categories LH/B and Sh(B), which assigns to each continuous map f : X → Y in LH/B its direct image f! applied to the local sections of X, is related to the self-duality of B-HMB by the condition f! = (f ).