categories: Re: categories or graphs?

[Ronnie Brown <mas010@bangor.ac.uk>]

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

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