Re: Monoidal structure on graphs

["John G. Stell" <j.g.stell@cs.keele.ac.uk>]

A structure closely related to the one which Francois Lamarche
asked about appeared in my thesis (1992) [see below] as a simple example of
a sesqui-category which is not a 2-category. I too would expect
it's appeared elsewhere, but I don't know where.

John Stell

\subsubsection{An Example of a Sesqui-Category}
We include an example to show that there are naturally occurring
sesqui-categories other than in connection with modelling
term rewriting. The underlying 
category is {\bf Graph}. Suppose there are graphs $G$ and $H$, and
graph morphisms $g,h : G \rightarrow H$. In this situation, the 2-cells
$\alpha : g \rightarrow h$ are assignments to each node $n$ of $G$ of a
path of edges from $ng$ to $nh$ in $H$. 

The compositions $\circ_R$ and 
$\circ_L$ are readily defined. If $f : F \rightarrow G$ then 
$f \circ_R \alpha$ assigns to a node $m$ of $F$ the path $(mf)\alpha$ in $H$.
For the left composition, suppose we have $k : H \rightarrow K$.
Since $n\alpha$ is a path in $H$, we obtain a path $(n\alpha)k$ by applying
$k$ to each of the edges in the path $n \alpha$. Thus we define 
$\alpha \circ_L k$ to be the assignment to $n$ of the path $(n \alpha)k$.

The vertical composition is the usual concatenation of paths. The identity
2-cells are assignments of zero length paths.