A Reduction Theorem for the Kripke-Joyal Semantics: Forcing Over an Arbitrary Category can Always be Replaced by Forcing Over a Complete Heyting Algebra

It is assumed that a Kripke-Joyal semantics A = 〈C,Cov,F, 〉 has been defined for a first-order language L. To transform C into a Heyting algebra C on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of C. A pretopology Cov is defined on C using the pretopology on C. A sheaf F is made up of sections of F that obey functoriality. A forcing relation is defined and it is shown that A = 〈 C,Cov,F, 〉 is a Kripke-Joyal semantics that faithfully preserves the notion of forcing of A. That is to say, an object a of COb forces a sentence with respect to A if and only if the maximal a-crible forces it with respect to A. This reduces a Kripke-Joyal semantics defined over an arbitrary site to a Kripke-Joyal semantics defined over a site which is based on a complete Heyting algebra. We will begin by recapitulating the definition of the Kripke-Joyal semantics since the details of the definition will be needed to prove the reduction theorem in the second part of this paper. 1. The Kripke-Joyal Semantics Robert Goldblatt’s version of a site from [1] and A. Kock and G.E. Reyes notion of forcing over a site [2] are used to establish the definition of the Kripke-Joyal semantics used in this paper. Alternative expositions of forcing in categorical contexts can be found in [3], [4], [5], and [6]. In this paper, except for set membership and inclusion, all compositions of arrows are written in the order of composition. Also, a category is considered to be small if its collection of arrows is a set. This allows us to form the first definition. Definition 1.1. A stack or presheaf of sets over a small category C is a contravariant functor C F −→ S where S is the category of sets. The functor © 2013 Imants Barušs and Robert Woodrow 2 Imants Barušs and Robert Woodrow category SC op is the category of all stacks over C. For a ∈ COb, s ∈ aF is called a germ. If A is a set then AP is its power set. The capital letters, I, X, Y , and so on, represent index sets. Definition 1.2. A pretopology on a small category C is an assignment COb Cov −→ ((CAr)P)P which takes a ∈ COb to a collection of sets of arrows in C with codomain a satisfying the following conditions: (i) the empty set φ 6 ∈aCov. (ii) the singleton { a 1a −→ } ∈ aCov. (iii) if { ax fx −→ a|x ∈ X } ∈ aCov and for each x ∈ X { ay f y −→ ax|y ∈ Yx } ∈ axCov,then { ay f y fx −→ a|y ∈ Yx and x ∈ X } ∈ aCov. (iv) if { ax fx −→ a|x ∈ X } ∈ aCov and b g −→ a ∈ CAr then for each x ∈ X the pullback b×a ax f ′ x −→ b of fx along g