Continuous and smooth envelopes of topological algebras

Since the time when the first optical instruments have been invented, an idea that the visible image of an object under observation depends on tools of observation became commonly assumed in physics. A way to formalize it in mathematics is the construction that assigns to an arbitrary object $A$ in a category $K$ its envelope ${\text{Env}}_{\Phi }^{\Omega }A$ in a given class of morphisms (a class of representations) $\Omega$ with respect to a given class of morphisms (a class of observation tools) $\Phi$. It turns out that if we take a sufficiently wide category of topological algebras as $K$, then each choice of the classes $\Omega$ and $\Phi$ defines a “projection of functional analysis into geometry”, and the standard “geometric disciplines”, like complex geometry, differential geometry and topology, become special cases of this construction. This gives a formal scheme of “categorical construction of geometries” with many interesting applications, in particular, “geometric generalizations of the Pontryagin duality” (to the classes of non-commutative groups). In this paper we describe this scheme in topology and in differential geometry.