Subject: categories: Terminology question wrt fibrations of categories.

From: "Ronald Brown" <[email-masked]>
Date: Tue, 6 Dec 2005 18:06:40 -0000

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

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From: "Prof. Peter Johnstone" <[email-masked]>
Date: Wed, 7 Dec 2005 09:05:10 +0000 (GMT)

> \phi is prone, supine, instead of cartesian, cocartesian
`Prone' and `supine' were invented by Paul Taylor; I copied them from him,
not the other way round. I'm sorry Ronnie doesn't like them; they seem to
me a very neat way of finding two words that both mean `lying
horizontally' but have an opposite handedness about them.

Peter Johnstone

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From: "Ronald Brown" <[email-masked]>
Date: Wed, 7 Dec 2005 18:21:52 -0000

Thanks for these comments.

I was mainly investigating the acceptance of these terms by the categorical
community, to decide if I should change for the new edition of my old book.

I agree with Paul's comments to me personally that it is a good idea to
avoid overworked terms (like `universal').

The issue is that for a morphism f: G \to H of groupoids, the notion of
quotient introduced by Philip Higgins, namely if f is full and Ob(f) is
surjective, is fine. The other important notion is that f comes from an
identification of objects, which in Paul's terminology would be supine
(w.r.t. the opfibration Ob: Gpds \to Sets). More vivid would be H is a
0-identification of G, that is the groupoid H is then obtained from G by an
identification of objects. It would tie in with other situations to say that
H is induced from G by Ob(f).

There is a good case for not introducing new words.

Ronnie