The fibrational formulation of intuitionistic predicate logic I: completeness according to Gödel, Kripke, and Läuchli, Part 1

Following thepattern of Lawvere's notion of hyperdoctrine, we single out certain classes of fibrations and use them, in the present paper and its sequel, to give an algebraic framework for the proof theory of intuitionistic predicate calculus. The two papers are organized around representation theorems that correspond to and strengthen the completeness theorems of the title. The present first part deals with the fibrational analog of GodeΓs completeness theorem and gives fibrational liftings of well-known categorical constructions. The present first part is a preparation for the main results to be given in the second part. 0 Introduction This is an introduction to both the present first part and the second part of the paper (cf. Makkai [18]). The numbering of the sections of the second part continues that of the first part; Sections 1 to 3 form the first part, Sections 4 to 6 the second part. By the "fibrational formulation of predicate logic" I mean the approach to predicate logic using the notion of hyperdoctrine, originally introduced by Lawvere in [15],[16], and its variants. Although Lawvere, and other authors following him (see below), used pseudofunctors as the basic ingredients, in this paper I adopt the essentially equivalent language of fibrations; hence the reference to fibrations in the title. A fibration is given by two categories B and C, the base category and the total category, respectively, of the fibration, together with a functor C -> B\ certain conditions have to be satisfied by these data for them to form a fibration; at this point, it is not important to know what these conditions are. In the fibrational formulation of logic, a(n axiomatic) theory is construed as a fibration. This is in contrast to the more widely discussed categorical approach to logic in which a theory is made into a (single) category (with appropriate properties). From the point of view of the present paper, the reason for adopting the more elaborate approach of fibrations is that it has the capability of incorporating the notion of proof (formal deduction), and not just Received August 8, 1989; revised February 28, 1991 FΊBRAΉONAL FORMULATION OF INTUΓΠONISΉC LOGIC 335 provability, which is what is grasped by the "one-category" approach. In brief, the ίibrational formulation gives a categorical proof theory. As was said above, Lawvere introduced the notion of hyperdoctrine in [15] and further discussed it in [16]. The original concept is made to suit higher order logic; it has "too much structure" for the purposes of first order logic. In Seely [23], a "first order" version of hyperdoctrine is attributed to Bέnabou [1] (a source not available to me). Seely [23],[24] uses a notion of "first-order" hyperdoctrine, closely related to the one I use in this paper, to build structures of proofs for theories in first order intuitionistic logic; his is the first systematic study of categorical proof theory for predicate logic. A version of the notion of hyperdoctrine, essentially identical to the one used in this paper although formulated only for posetal fibers, is given and called "first order hyperdoctrine" by Pitts [21]. The (main) version in this paper, called Heyting fibration, or more neutrally h-fibration, was arrived at by considering exactly what one needs to model the structure of proofs in an arbitrary theory in first order intuitionistic logic. The categorical approach to the proof theory of propositional logic is due to Lambek; his work relates the proof theory of a range of propositional calculi with concepts of structured category. References are [10],[ll],[12], and the recent book with Scott, [13]. Szabo's work [25] on the algebra of proofs is a thorough investigation of several propositional calculi from a categorical point of view. Whereas the Lambek-style proof theory for propositional logic requires (single) structured categories (e.g., cartesian closed categories), for the proof theory of predicate logic the more elaborate structures of fibrations are needed. Lawvere pointed to the possibility of the uses made in this paper of the concept of hyperdoctrine and its variants already in [16] as the following quotations show: "Thus, this hyperdoctrine [^(Set) in the present paper; see Section 1] may be viewed as a kind of set-theoretical surrogate of proof theory (honest proof theory would presumably also yield a hyperdoctrine with non-trivial [non-posetal] P(X) [fibers], but a syntactically presented one). " "It appears that abstract structures of this kind [hyperdoctrines] are also intimately related to . Lauchli's complete sematics for intuitionistic logic [the work in Lauchli [14]], although the precise relationship is yet to be worked out. " In fact, the main result of the second part of the present paper [18], Theorem 6.1, a representation theorem for Heyting fibrations, was inspired by Lauchli's work mentioned by Lawvere, and it is intended as a generalized and strengthened version of Lauchli's result. In (the two parts of) this paper, an abstract theory of the relevant types of fibration is given, giving results of a representation theoretic nature; in the sequel (Makkai [19]), I will discuss in detail the connections of these concepts to the usual concepts of proof theory. Tb help understand the motivation for this paper, I will now give an informal description of how the fibrational formulation works. If G ί i s a fibration, Λ is an object in 2*, then S A denotes the fiber of G over A: the subcategory of C with objects and arrows that map to A and 1Λ respectively. Given a theory in possibly many-sorted intuitionistic predicate logic, we form a fibration along the following lines. The arrows of the base category are given, in a natural way, by the terms of the theory; the objects are formal products of sorts. The only