A canonical model for presheaf semantics

Presheaf models [Osius, 1975, Fourman and Scott, 1979, Moerdijk and Mac Lane, 1992] are a simple generalization of Kripke models and provide one of the most natural semantics for intuitionistic predicate logic. Although a completeness result for this semantics is known, this is usually obtained either non-constructively, by defining a canonical Kripke model, or indirectly, e.g. via categorical equivalences with Ω−models as in [Troelstra and Van Dalen, 1988]. This paper describes an elementary canonical model construction by means of which completeness is established in a perspicuous, direct and constructive way. Presheaf semantics. We start by recalling the definition of presheaf semantics for intuitionistic logic. Definition 1 (Presheaf models for intuitionistic logic). Let L be a predicate logic language with equality. A presheaf model for L consist of a presheaf of firstorder structures for L over a Grothendieck site (C, ). That is, to any object u of C we associate an L−structure Mu and to any f : v → u we associate a homomorphism f : Mu →Mv in a functorial way. The domain of Mu is denoted |Mu|, while the interpretation of a function symbol f and of a relation symbol R in Mu are denoted fu, Ru respectively. In addition, we require our structure to satisfy separateness and an analogous property asserting the local character of atomic formulas. Separateness For any elements a, b of Mu, if u {fi : ui → u | i ∈ I} and for any i ∈ I we have a fi= b fi , then a = b. Local character of atoms For any n−ary relation symbol R and any tuple (a1, . . . , an) from Mu, if u {fi : ui → u | i ∈ I} and for any i ∈ I we have (a1 fi , . . . , an fi) ∈ Rui , then (a1, . . . , an) ∈ Ru. Any assignment ν into a structure |M | extends in the usual way to an interpretation [t]ν of any term t into M . Moreover, if in a presheaf model we have f : v → u and an assignment ν into Mu, then an assignment f ◦ν into Mv is naturally induced. We will allow ourselves to write again ν for the resulting assignment. Definition 2 (Forcing on a presheaf model). Formulas of L can be interpreted on an object u of a given presheaf model M , relative to an assignment ν into Mu as follows. in ria -0 06 18 86 2, v er si on 1 3 Se p 20 11 Author manuscript, published in "Topology, Algebra and Categories in Logic (TACL) 2011 (2011)"