Sheaves and presheaves with topological algebraic structures

Abstract

The purpose of the subsequent discussion is to show that, apart from the classical framework of differential (C^∞-)manifolds (generalized or not), as exposed in the preceding chapter, certain types of topological algebras too furnish further non-trivial examples, where one can apply the point of view of the present treatment, getting thus a considerable part of the standard machinery of the classical differential geometry. In this context, we still notice that, here too, the key-idea is to replace the standard sheaf of germs of (C-valued) smooth functions, as appears in the classical theory (see e.g. Chapt. X; (1.3)), through the sheaf of germs of (continuous) sections of a suitable topological algebra sheaf, the aforementioned one of the classical theory constituting, in effect, simply a very special (important, nevertheless!) example of the latter type of sheaf; yet, one further seeks for a de Rham complex (generalized or not) analogous to the one already considered in the preceding (see, for instance, (1.8) or even (1.10) of the previous Chapter).