Category inheritance in going from static knowledge systems to dynamic ones

The value of invoking the apparatus of the theory of categories for abstract knowledge description in a product system was previously demonstrated by the authors. The apparatus was initially intended for the description of the common, i.e., static expert systems that have the widest circulation in artificial intelligence and related fields. In such a case, the so-called dynamic expert systems, which allow the changing of data and knowledge during operation appeared to be outside to some extent. As the dynamic expert system was based on the static ones in the schemes suggested previously, it is reasonable that the question arises of the circulation of the category-theoretic apparatus of these systems. To do this required a certain generalization of the category-theoretic apparatus that was developed previously, related to the introduction of the so-called fibred categories and corresponding context lattices. The whole complex of problems that arise in this connection is considered in this paper.