Adjointness in Foundations

In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globalized into functors, such as Spec, carrying much more information. Also, “theories” (even when presented symbolically) are viewed explicitly as categories; so are the background universes of sets that serve as the recipients for models. (Models themselves are functors, hence preserve the fundamental operation of substitution/composition in terms of which the other logical operations can be characterized as local adjoints.) My 1963 observation (referred to by Eilenberg and Kelly in La Jolla, 1965), that cartesian closed categories serve as a common abstraction of type theory and propositional logic, permits an invariant algebraic treatment of the essential problem of proof theory, though most of the later work by proof theorists still relies on presentation-dependent formulations. This article sums up a stage of the development of the relationship between category theory and proof theory. (For more details see Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), pp. 1–14, and Marcel Dekker, Lecture Notes in Pure and Applied Mathematics, no. 180 (1996), pp. 181–189.) The main problem addressed by proof theory arises from the existential quantifier in “there exists a proof. . . ”. The strategy to interpret proofs themselves as structures had been discussed by Kreisel; however, the influential “realizers” of Kleene are not yet the usual mathematical sort of structures. Inspired by Lauchli’s 1967 success in finding a completeness theorem for Heyting predicate calculus lurking in the category of ordinary permutations, I presented, at the 1967 AMS Los Angeles Symposium on Set Theory, a common functorization of several geometrical structures, including such proof-theoretic structures. As Hyperdoctrines, those structures are described in the Proceedings of the