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Topological universes and smooth Gelfand-Naimark duality. (English) Zbl 0548.46054

Mathematical applications of category theoy, Proc. Spec. Sess. 89th Annu. Meet. Am. Math. Soc., Denver/Colo. 1983, Contemp. Math. 30, 244-276 (1984).
[For the entire collection see Zbl 0534.00008.]
According to a theorem of J. Boman a mapping \(f:{\mathbb{R}}^ n\to {\mathbb{R}}\) is smooth if \(f\circ g\) is smooth for every smooth curve \(g:{\mathbb{R}}\to {\mathbb{R}}^ n.\) So it is possible to recover all smooth maps \(f:{\mathbb{R}}^ n\to {\mathbb{R}}^ k\) if one knows all smooth maps from \({\mathbb{R}}\) to \({\mathbb{R}}\) as well as all smooth curves in \({\mathbb{R}}^ n\). In the light of this result several authors set up a differential calculus by starting e.g. with the notion of a smooth curve (which is often given in a natural way, so for a mapping from \({\mathbb{R}}\) into a topological vector space). Then they extend this notion in such a way that Boman’s theorem becomes true in the new setup. In the paper under review it is shown that these procedures can be regarded as special cases of a general categorical construction: Starting with a category which is in general not very large (e.g. all smooth curves in \({\mathbb{R}}\) or all smooth maps between finite- dimensional convex sets) he constructs a topological category which satisfies one more axiom and is called a topological universe. A slight generalization of this gives the notion of a topological universe over a topological universe and in this way one gets many interesting categories: On one hand the category \({\mathcal L}iss\) of lissome spaces, whose morphisms can be regarded as ”smooth” maps or categories whose morphisms can be seen as some kind of differentiable maps. But on the other hand he also gets the categories of bornological spaces or that of simplicial spaces.
It was proved by E. Binz that two c-embedded convergence spaces X and Y are homeomorphic if and only if there is a bicontinuous \({\mathbb{R}}\)- algebra-isomorphism between C(X,\({\mathbb{R}})\) and C(Y,\({\mathbb{R}})\), where the function spaces carry the continuous convergence structure. In the second part of the paper the author shows that an analogue of this theory holds in every topological universe over the category of convergence spaces which contains an object whose underlying convergence space is that of the reals and which satisfies some more mild conditions.
The reviewer would like to point out that this paper, although categorical in nature, is very readable for non-experts in category theory, since all the relevant notions are clearly explained.
Reviewer: H.-P.Butzmann

MSC:

46M15 Categories, functors in functional analysis
46G05 Derivatives of functions in infinite-dimensional spaces
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
46A99 Topological linear spaces and related structures
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
46M40 Inductive and projective limits in functional analysis

Citations:

Zbl 0534.00008