Higher order symmetry of graphs

1 Symmetry in analogues of set theory This lecture gives background to and results of work of my student John Shrimpton [19, 20, 21]. It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein's famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. The structure of group alone may not give all the expression one needs of the intuitive idea of symmetry. One often needs structured groups (for example topological, Lie, algebraic, order, ...). Here we consider groups with the additional structure of directed graph, which we abbreviate to graph. This type of structure appears in [18, 14]. We shall associate with a graph A a group AUT(A) which is also a graph. The vertices of AUT(A) are the automorphism of the graph A and the edges between automorphisms give an expression of " adjacency " of automorphisms. The vertices of this graph form a group, and so also do the edges. The automorphisms of A adjacent to the identity will be called the inner automorphisms of the graph A. One aspect of the problem is to describe these inner automorphisms in terms of the internal structure of the graph A. The second theme is that of regarding the usual category of sets and mappings as but one environment for doing mathematics, and one which may be replaced by others. We use the word " environment " here rather than " foundation " , because the former word implies a more relativistic approach. The other environment we choose here is the category of directed graphs and their morphisms. We define this category, and then use methods analogous to those of set theory within this category. * This paper is an account of a lecture " Groups which are graphs (and vice versa!) " given to the Fifth September Meeting of the Irish Mathematical Society at Waterford Regional Technical College, 1992. It was published as [4].