Terminology question wrt fibrations of categories.
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
On Tue, 6 Dec 2005, Ronald Brown wrote:
on the other hand Paul Taylor, following Peter Johnstone, I understand, uses \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
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 ----- Original Message ----- From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk> To: "Ronald Brown" <ronnie@ll319dg.fsnet.co.uk> Cc: <categories@mta.ca> Sent: Wednesday, December 07, 2005 9:05 AM Subject: Re: categories: Terminology question wrt fibrations of categories.
On Tue, 6 Dec 2005, Ronald Brown wrote:
on the other hand Paul Taylor, following Peter Johnstone, I understand, uses \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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Prof. Peter Johnstone -
Ronald Brown