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Manifolds of continuous structures. (English. Russian original) Zbl 1159.58001

J. Math. Sci., New York 154, No. 4, 523-538 (2008); translation from Sovrem. Mat., Fundam. Napravl. 23, 71-86 (2007).
The theory of categories and functors replaces collections of sets considered as individual sets with some given structures, and is the explicit inclusion of its objects as well as transformations admitted by the structure of the objects, namely, such correspondences from one set into others that do not violate their structure.
In this paper, the authors describe systems by categories. The system itself is a category that consists of a class of objects and a class of morphisms. For example, a system for transforming in whose states arbitrary correspondences are admissible will be distinguished from a system where the same sets are transformed only in a one-to-one manner. Processes occurring in the first system are richer than those in the second one; transitions between states with a variable number of elements are admissible in it, while in the second system the number of elements in different states is equal. For morphism, like preimages, the so-called “multiplication” that coincides with the composition of mappings is defined.
The authors study in details duality between structures of algebraic type and structures of continuous type, and continuous structures over the category of sets and their manifolds.

MSC:

58A05 Differentiable manifolds, foundations
18A99 General theory of categories and functors
57R99 Differential topology
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