A convenient setting for differential calculus

Abstract

Cartesian closed categories play an important r6le in many aspects of mathematics. They appear in algebraic geometry, in logic, in topology. Quite naturally one is tempted to use this notion also for differential calculus. First attempts in this direction were undertaken by A. Bastiani [l] and then by A. Frolicher and W. Bucher [4]. They all used the notion of limit spaces for their generalizations of differential calculus. In retrospect one may say that A. Bastiani used a “good” definition of differentiability but the category she chose did not allow Cartesian closedness. A. Frolicher and W. Bucher took Cartesian closedness exactly as their goal. But because of their “bad” definition of continuous differentiability, their special types of limit vector spaces became increasingly complicated. Because questions of continuity and differentiability are local and topological spaces have by definition a local structure, it has many advantages to establish differential calculus in a “pure” topological setting. And then the question is: Can a differential calculus be so established in a topological setting as to obtain Cartesian closedness in the infinitely often differentiable case? For a long time it even seemed impossible to obtain anything like Cartesian closedness for continuous maps not to speak of differentiable ones. But in 1963 Gabriel and Zisman proved [6] that the category V2? of compactly generated hausdorff spaces is Cartesian closed and this is a full subcategory of the category of topological spaces. Starting from this category ‘%%, we are quite naturally led to investigate the category %%V of compactly generated vector spaces. Observing that the HahnBanach theorem is the tool for the proof of the so-called mean value theorem of differential calculus, we see that not all compactly generated vector spaces are convenient for a differential calculus. But there is a nice full subcategory of VYV and the objects of this subcategory will provide our convenient setting.