Some groups related to the symmetry of a directed graph (Q1176699)

In studying the automorphism group \(\Aut(G)\) of a group \(G\) it is useful to consider inner automorphism maps \(G\to\Aut(G)\) defined by \(x\mapsto gxg^{-1}\). The inner automorphism map with the action of \(\Aut(G)\) on \(G\) has the structure known as a crossed module. The author associates a crossed module to the automorphism group of a directed graph by embedding the automorphism group in a bigger structure which is both a group and a directed graph. He uses the theory of cartesian closed categories and monoidal closed structures. He obtains interesting analogies with group theory in a category of directed graphs. Of course, a transfer of these results to the category of undirected graphs will need other methods since the category of undirected graphs is not cartesian closed.