Let Cat be the 2-category of small categories. Functors F: C D can be thought of as morphisms of Cat. Full and faithfull embedding of categories

X, Y Ob C (12)

![0_image_0.png](0_image_0.png)

HomC (X, Y) HomD (FX, FY) (13)
Natural transformation. Covariant and contravariant functors. Forgetfull functor. Representable functors. Pre-Sheaves and Sheave

(X, OX) *U X , OX(U)* (14)

![0_image_1.png](0_image_1.png)

![0_image_2.png](0_image_2.png)

 Categories of functors or Functor categories (morphisms in this category are natural transformations between functors), AnnkE, **AnnkE** *(F,G)*. K-functor. K-tensor functor. 

## 8 Ontological Knowledge Bases

 By ontological knowledge base (KB) we will understand the KB that consist of in some way organized ontologies. In the full version of the paper we will present elements of the KB.

E-Learning lessons of the course Foundations of Research Activities are designed to guide students through information about the history and facts related to a research activities in the problem domain Electromagnetism and Gravitation and to help students perform specific tasks in the framework of computer-assisted methods and technologies to justification of conjectures (hypotheses). The cource is based on books and paper by Feynman R, Leighton R, and Sands M. (Feynman, Leighton, Sands, 1989), Landau L. D. and Lifshitz E. M. (Landau, Lifshitz, 1995), Hawking S., Ellis G., Landshoff P., Nelson D., Sciama D., Weinberg S ( Hawking, Ellis, Landshoff, Nelson, Sciama, Weinberg, 1975) and on Wikipedia (Wikipedia, 2009). E-Learning lessons of the course Virtual Modeling of Complex Systems are designed to guide students through information about the history and facts related to virtual modeling of complex systems in the problem domain Virtual Reality and to help students perform specific tasks under designing elements of virtual reality (elements of virtual classrooms are included). E-Learning lessons of the course Introduction to String Theory are designed to guide students through information about the history and facts related to elements of closed and open strings (Polchinski, 1998) (Wess and Bagger, 1983) (Glazunov, 2008a, 2008b) with applications to the problem domain Electromagnetism and Gravitation and to help students perform specific tasks in the framework of closed and open strings.

## 9 Discussion And Conclusions

In the paper, we have discussed applications of formal methods, algebra and category theory to design of elements of ontologies and ontological knowledge bases of electronic courses. We have illustrated our approach on the example of the design of elements of ontologies and ontological knowledge bases of next three courses: Foundations of Research Activities, Virtual Modeling of Complex Systems and Introduction to StringTheory.

The next step towards a Knowledge Based Systems Theory of e-learning is a more complete understanding of the category theory basics of problem_solving/formal_methods