categories: Re: categories or graphs?

[S Vickers <s.j.vickers@open.ac.uk>]

Peter is quite right. I was mistaken in saying graph shape and category
shape are mathematically equivalent. Category shape for diagrams has graph
shape plus the commutativities, which graph shape on its own does not
capture. His justification for considering category shape is much more
profound than the ones I mentioned.

Nonetheless, when verifying the conditions for cones/cocones, or for
limits/colimits, the commutativities make no difference and it suffices to
check on a generating graph. Thus it is worth bearing in mind that in such
contexts one might correctly and more simply just use graph shape, as
Gillian and I did in "Presheaves as configured specifications". (For those
interested it's on the web via my home page at http://mcs.open.ac.uk/sjv22 .)

Steve Vickers.