categories: Re: categories or graphs?

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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