Diagonal arguments and cartesian closed categories
In May 1967 I had suggested in my Chicago lectures certain applications of category theory to smooth geometry and dynamics, reviving a direct approach to function spaces and therefore to functionals. Making that suggestion more explicit led later to elementary topos theory as well as to the line of research now known as synthetic differential geometry. The fuller development of those subjects turned out to involve a truth value object that classifies subobjects, but in the present paper (presented in the 1968 Battelle conference in Seattle) I refer only to weak properties of such an object; it is the other axiom, cartesian closure, that plays the central role. Daniel Kan had recognized that the function space construction for simplicial sets and other categories is a right adjoint, thus unique. Because this uniqueness property of adjoints implies their main calculational rules, I took the further axiomatic step of defining functor categories as a right adjoint to the finite product construction in my 1963 thesis. In 1965, Eilenberg and Kelly introduced the term closed to mean that there is a hom functor valued in the category itself. Such a hom functor is characterized in a relative way as right adjoint to a given tensor product functor; we concentrate here on the absolute case where the tensor is cartesian. Although the cartesian-closed view of function spaces and functionals was intuitively obvious in all but name to Volterra and Hurewicz (and implicitly to Bernoulli), it has counterexamples within the rigid framework advocated by Dieudonné and others. According to that framework the only acceptable fundamental structure for expressing the cohesiveness of space is a contravariant algebra of open sets or possibly of functions. Even though such algebras are of course extremely important invariants, their nature is better seen as a consequence of the covariant geometry of figures. Specific cases of this determining role of figures were obvious in the work of Kan and in the popularizations of Hurewicz’s k-spaces by Kelley, Brown, Spanier, and Steenrod, but in the present paper I made this role a matter of principle: the Yoneda embedding was shown to preserve