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 23-Sep-2001 13:37:39 -0300,3348;000000000000-00000021