From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1076 Path: news.gmane.org!not-for-mail From: "John G. Stell" Newsgroups: gmane.science.mathematics.categories Subject: Re: Monoidal structure on graphs Date: Thu, 18 Mar 1999 10:55:08 +0000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017551 29402 80.91.229.2 (29 Apr 2009 15:05:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:05:51 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Thu Mar 18 12:44:20 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id IAA18822 for categories-list; Thu, 18 Mar 1999 08:48:14 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Content-MD5: rUxkjrwDZJdVdKTrk48DUA== Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 29 Xref: news.gmane.org gmane.science.mathematics.categories:1076 Archived-At: 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.