nLab dagger-graph

Idea

In view of the relation between categories with their underlying (reflexive) directed graphs and in view of the notion of dagger-categories it makes sense to say that:

A $\dagger$-graph is a (reflexive) directed graph equipped for each pair of vertices $x,y$ with an involution $\sigma_{x,y} \colon V(x,y) \to V(y,x)$ on sets of edges between these vertices, changing the direction.

This is one way to think of undirected graphs as directed graphs with extra structure. In fact, some authors define pseudographs to be directed graphs equipped with such an involution.

The point of the terminology here is that the underlying graph of any dagger-category is canonically a dagger-graph, in the above sense.

Related concepts

• pseudograph