Terminology question wrt fibrations of categories.

["Ronald Brown" <ronnie@ll319dg.fsnet.co.uk>]

I am writing about matters to do with computation of colimits of a category
X in terms of colimits of a category B when there is a bifibration P: X -->
B.

Terminology already in use is

P is cartesian
P is cocartesian
a lifting of u in B to \phi in X may be cartesian, cocartesian

on the other hand Paul Taylor, following Peter Johnstone, I understand, uses
\phi is prone, supine, instead of cartesian, cocartesian

For the cofibration (?opfibration?) Ob: Groupoids  --> Sets, Philip Higgins
(1971) and I (1968) have previously used  `universal' for cocartesian. In
this situation, I would be happier with say 0-final instead of universal.
But `supine' does not ring a bell with me, and carries  a  pejorative tone.

Maybe for the general situation P: X --> B we could use
P-initial, P-final morphism in X
for cartesian, cocartesian  morphism
which would at least carry some intuition as to the meaning. Comments?

I need to make a decision soon for the revision of my old topology book. Not
much will be changed, and I might leave the old terminology and refer to
more modern uses. However for the book on Nonabelian algebraic topology, I
really do need to use modern terminologym, whatever that is, so it would be
best to be consistent.

I have been looking at Thomas Streicher's notes on fibrations, and at Paul
Taylor's Practical Foundations.

For my interest, see slides of a recent seminar at Oxford

www.bangor.ac.uk/r.brown/oxford2811105.pdf

called `Induced constructions and their computation'.

Ronnie Brown
www.bangor.ac.uk/r.brown