On a categorical approach to the theory of universal $C^*$-algebras

The report is devoted to the development of a categorical approach to the theory of universal $C^*$-algebras. This approach was proposed by Loring in 2010. Within the framework of this approach, categories of representations satisfying a number of natural axioms are considered. Such categories are called $C^*$-relations. In the case when a $C^*$-relation defines a universal $C^*$-algebra, it is called compact. We introduce $*$-polynomial relations and study their connection with compact $C^*$-relations. A criterion for the existence of universal $C^*$-algebras is obtained. Further, we discuss a characterization of functors between compact $C^*$-relations in terms of $*$-polynomial relations. The concept of the soft image of a functor is introduced. We use this concept for describing properties of factorization functors.