It may be helpful to remark that in Ronald Brown and Anne Heyworth, `Using rewriting systems to compute left Kan extensions and induced actions of categories', J. Symbolic Computation 29 (2000) 5-31. we introduce the notion of a \textbf{Kan extension presentation}. This is a quintuple $\mathcal{P}:=kan\lan \Gamma|\Delta|RelB|X|F \ran$ where \begin{enumerate}[i)] \item $\Gamma$ and $\Delta$ are graphs, \item $cat\lan \Delta | RelB \ran$ is a category presentation, \item $X: \Gamma \to U \sets$ is a graph morphism, \item $F: \Gamma \to U P\Delta$ is a graph morphism. \end{enumerate} The idea is analogous to a presentation of a group, where one gives a hopefully finite amount of information in order to compute, in some sense and in some cases, the group or in this case a Kan extension. Ronnie Brown "Dr. P.T. Johnstone" wrote:
On Fri, 21 Sep 2001, S Vickers wrote:
I haven't got either books in front of me at the moment, so I hope I'm not going off on a tangent. However, there is a definite choice of approach here: Is the shape of a diagram a graph or a category?
They are mathematically equivalent. If a graph-shaped diagram has shape A, then one can form the free category Path(A) over A (objects are the nodes, morphisms are chains of edges) and uniquely extend the graph morphism from A to a functor from Path(A).
I guess the reason for choosing the category-shaped diagrams is that one can then apply directly all that is known about functors and natural transformations.
However, that choice is not entirely benign. For a start, it seems beyond doubt that when one draws a diagram one is drawing a graph. The graph is easier to deal with mentally, and a finite graph may generate an infinite category.
No, that's not the reason. Steve is right that what we actually draw and call "diagrams" are the images of graph morphisms, but we also make assertions (often without stating them explicitly) that certain parts of the diagrams commute, so that what we think of as the "shape" of a diagram is not simply a directed graph but (a presentation of) a category. For example, if I want to talk (as I often do) about properties of reflexive coequalizers in a category, I need to consider diagrams whose shape is the category generated by morphisms f: A --> B, g: A --> B and s: B --> A subject to the equations fs = gs = 1_B. If Steve is only willing to allow me to talk about diagrams whose shape is (the free category generated by) a directed graph, then I can't do this.
Peter Johnstone
-- Prof R. Brown, School of Informatics, Mathematics Division, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382681 fax: +44 1248 361429 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ (Links to survey articles: Higher dimensional group theory Groupoids and crossed objects in algebraic topology) Raising Public Awareness of Mathematics CDRom Version 1.1 http://www.bangor.ac.uk/~mas010/CDadvert.html Symbolic Sculpture and Mathematics: http://www.cpm.informatics.bangor.ac.uk/sculmath/ Centre for the Popularisation of Mathematics http://www.cpm.informatics.bangor.ac.uk/ 26-Sep-2001 16:29:14 -0300,968;000000000000-00000028