Hyperdoctrines, Natural Deduction and the Beck Condition

In the late sixties F. W. LAWVERE showed that the logical connectives and quantifiers were examples of the categorical notion of adjointness. In [9] and [lo] he amplified this notion by a more thorough discussion of the structure of a hyperdoctrine, which had much of the flavour of intuitionistic logic with equality. In this eontext it was natural to “stratify” formulae and proofs according to the free variables occurring in them, a procedure later to become standard in categorical logic. (See MAKKAIREYES [12], FOURMAN [l], KOCK-REYES [7], for example.) In this paper, we make the relationship between hyperdoctrines and logic precise, showing that hyperdoctrines are naturally equivalent to first order intuitionistic theories with equality, where here “theory” is intended to include some proof theoretic structure, and not merely the notion of entailment. Moreover, we will show that this equivalence restricts to one giving a natural logical interpretation to the BECK (or CHEVALLEY) condition: in a given hyperdoctrine, the Beck condition for a pullback diagram is just the condition that the corresponding theory “recognizes ” the pull back.