Categories and foundations of the theory of nonlinear controlled dynamical systems. I (Q1811507)

According to the introduction, ``the exposition of foundations of a particular mathematical theory begins with the definition of\dots notions.'' In the first part the author develops ``an approach that permits one to use category theory and thus introduce the above mentioned notions in a very natural manner''. In fact, he introduces only a few definitions of category theory. Moreover, the author only considers categories whose objects are sets. The reviewer's opinion: the same insights can also be obtained without using objects and morphisms. In the second chapter, which requires more than half of the paper, the author introduces basic notions of foundations of differential geometry. References to some text books would have been better, for the introduction given here is not up to date. Two notions, however, seem to be new: ``affine distribution'' and ``t-characteristic codistribution''. But it is not obvious that the definition of the latter one is geometrically meaningful.