Speaker
Viljem Tisnikar
(University of Primorska)
Description
Since Euler's conceptualization of 7 bridges of Königsberg in 1735, Graph theory is nowadays a well established field of mathematics, especially applied. But if today one wants to apply mathematics for modeling all kinds of systems and processes, one has to go a step further and treat vertices, edges and paths in a graph more algebraically unified. A graph is also not an object per se, but communicates with other graphs, other mathematical objects, and with real world problems via certain morphisms, which also can be given unified characteristics. The language for such a unification is Category theory, a conceptual framework, allowing us to see the universal components of structures and of their interrelations.
Category theory initiated with a paper by Eilenberg & Mac Lane in 1945. The central notion was "natural transformation", and in order to give a general definition of it, the authors defined "functor" and "category". So, there was "no premeditated initiation of a unifying mathematical language" and it was also only later recognized that a "groupoid" (Brandt 1927) has a very simple definition in a categorical setting. The cornerstone of Category theory today is the concept of "adjoint functors" (Kan 1958). Applying the concept to a special kind of groupoids, one obtains an "adjoint equivalence" relation, which exhibits a remarkable level of symmetry between objects and morphisms involved, simultaneously.
In this talk I will give a brief overview of the concepts from previous paragraph through their relations with well known Graph theoretical concepts, and some words about a paradigm shift from Object-centric to Arrow-centric mathematics. The latter may be the starting point for a novel approach in the field of Knowledge Representation.
Primary author
Viljem Tisnikar
(University of Primorska)
