For the revision of my old topology book, I want to keep the analogies which the fibrations of categories concept brings out very well - I am coming (very late!) to the view that this is an important part of `categories for the working mathematician' (the concept, not necessarily the book). In the general topology part of the book, I have final topologies with respect to a function, and also identification maps, as in Bourbaki. So (unless anyone can quickly come with anything better) I have decided on replacing (a) `universal morphism of groupoids f: G \to H' as in the current text and Philip Higgins' book, and which uses an overused word `universal', by (a') ` f gives H the final structure with respect to G and Ob(f)' (b) `under these circumstances, f is a 0-identification morphism if also Ob(f) is surjective'. I initially (!) wanted to say `f is a 0-final morphism' instead of (a') but Tim pointed out it was initial in an appropriate category! Another possibility is `H has the induced structure w.r.t Ob(f)', and to use the res/ind terminology from representation theory and Mackey functors. Comments on these issues welcome. But the aim is to use terminology which has associations and emphasises analogies. The new title will be `Topology and groupoids', which seems better to reflect the content. It will (all being well) be available as a print and ebook, with the ebook in color and hyper-reference. Ronnie www.bangor.ac.uk/r.brown ----- Original Message ----- From: "Peter May" <may@math.uchicago.edu> To: <categories@mta.ca> Sent: Wednesday, December 21, 2005 2:16 AM Subject: categories: Right on, Jean!
It's funny, but exactly that question of terminology (prone, supine, etc) came up a few weeks ago in some joint work with a student here. I made the same case to him, almost exactly, that Jean Benabou just made. And one other objection: I'd like category theory no longer to be regarded as nonsense in this country --- it still is in many quarters, as I could easily prove --- and such terminology is not exactly helpful to the cause!
Peter May
ps: Then again, I don't much like using the overused words cartesian and cocartesian.