* Terminology re fibrations and opfibrations of categories
@ 2005-12-22 8:07 Ronald Brown
2005-12-26 21:57 ` Eduardo Dubuc
0 siblings, 1 reply; 3+ messages in thread
From: Ronald Brown @ 2005-12-22 8:07 UTC (permalink / raw)
To: categories
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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* Re: Terminology re fibrations and opfibrations of categories
2005-12-22 8:07 Terminology re fibrations and opfibrations of categories Ronald Brown
@ 2005-12-26 21:57 ` Eduardo Dubuc
0 siblings, 0 replies; 3+ messages in thread
From: Eduardo Dubuc @ 2005-12-26 21:57 UTC (permalink / raw)
To: categories
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)
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Terminology re fibrations and opfibrations of categories
@ 2006-01-02 16:20 Hans-E. Porst
0 siblings, 0 replies; 3+ messages in thread
From: Hans-E. Porst @ 2006-01-02 16:20 UTC (permalink / raw)
To: categories
Note that the suggestion below is standard terminology since about 30
years. See also
Adamek, Herrlich, Strecker: Abstract and concrete categories; Wiley 1990
(also available at http://katmat.math.uni-bremen.de)
H.-E. Porst
Am 26.12.2005 um 22:57 schrieb Eduardo Dubuc:
> 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"
--
Hans-E. Porst porst@uni-bremen.de
Bremen, Germany Fax: +49-421-75643
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