Hello,

I am rereading Walters' book in particular the functor chapter. I am also reading Barr & Wells "Category Theory for Computing Science" (erd edition) ... in particular chapter 4 on diagrams, sketches, etc. In Walters' functor chapter, I am wondering whether Example 13 is totally correct. It doesn't seem to me that "A" is a graph and what is called a functor is really a graph morphism or to put it another way, Example 13 should be couched in terms of a (formal!!) diagram of shape "A". I know that somebody will respond and say that "A" is really a category with implicit identity arrows. However, my retort would be that "A" is intended to be a graph (without composition). What do others think?

Regards, Bill Halchin

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. One particular context where I have found graph-shaped diagrams easier to handle mathematically is in cocompletion. Suppose you have a category C and want to adjoin, freely, all colimits, to give a bigger category Cocomp(C). One approach is to use diagrams as the objects of Cocomp(C), representing their colimits, and another is to use presheaves. The relationship between the two is best seen with graph-shaped diagrams, for they present presheaves in a simple way. Going to path categories just complicates everything, and covers up the simple correspondence between finitely shaped diagrams and finitely presentable presheaves (needed for finite cocompletion). This is discussed in my paper with Gillian Hill, "Presheaves as configured specifications", Formal Aspects of Computing 13 (Sep 2001) pp. 32-49. Steve Vickers. 21-Sep-2001 17:56:10 -0300,1535;000000000001-0000001e