Subject: categories: Right on, Jean!

From: Peter May <[email-masked]>
Date: Tue, 20 Dec 2005 20:16:24 -0600

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.

---

From: "Ronald Brown" <[email-masked]>
Date: Wed, 21 Dec 2005 18:08:20 -0000

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