Subject: categories: Terminology re fibrations and opfibrations of categories

From: "Ronald Brown" <[email-masked]>
Date: Thu, 22 Dec 2005 08:07:43 -0000

To add to my previous email, I'd like reactions to the following
terminology:

Let P: X \to B be a functor. A morphism u: x \to y in X is  cofinal w.r.t.
P, and y is the P-final object w,r,t u and P , if ... (and here we have the
usual notion of cocartesian).

Dually,  u is coinitial, and x is the initial object w.r.t   u and P if ...
(and here we have the usual notion of cartesian).

In situations where P is understood, we can then talk about cofinal and
coinitial morphisms, and structures or objects or (in my case, groupoids).

An advantage is that the direction of the notion and its dual should be
clear.

If f=P(u), I would then write \bar{f}: x \to f_*(x) in the first case, and
\underline{f}: f^*(y) \to y in the second. I would also call f_*(x) the
object induced by f.  What is a handy name for f^*(y)? The restriction of y
by f?

All these notions occur for modules, crossed modules, ...... and relate to
change of base.

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

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From: Eduardo Dubuc <[email-masked]>
Date: Mon, 26 Dec 2005 18:57:48 -0300 (ART)

Concerning  Ronnie wanderings about terminology around the word  FINAL,
the following is pertinent:

I am just writing a paper with Luis Espannol where we need to develop (the
basic part of the theory of cartesian and cocartesian arrows) for families

we use the following terminology:

consider a functor  U: C ---> S, then:

1)  a family in C              Z _i ---> X

over                           R_i --->  S   is    FINAL   iff:

given   S ---> T = UY  such that there exists   Z_i --->Y   over
R_i ---> S ---> T (that is,  R_i ---> S ---> T lifts), then there exists a
unique   X ---> Y over   S ---> T (that is,  S ---> T lifts).

For topological spaces this is the usual Bourbaki notion of final
topology.

When U is not understood, we call this  "U-FINAL"

Notice that for single arrows, we have (proved in the SGA on fibered
categories)

            Z ---> X is final     iff            it is cocartesian
                                          and cocartesian  arrows compose

2)  a family in C            Z _i ---> X

over                         R_i --->  S  is  SURJECTIVE   iff:

the family  R_i --->  S  is an strict (or regular) epimorphic family in S

Our aim is to prove under some natural and minimal assumptions:

  Z _i ---> X   is  strict epimorphic     iff    it is  final  surjective

All this is already done

Here the leading examples are the topological spaces  and  the
quasitopological spaces in the sense of Spanier  (and the whole theory of
concrete quasitopoi over S = Sets)