Terminology re fibrations and opfibrations of 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
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)
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
participants (3)
-
Eduardo Dubuc -
Hans-E. Porst -
Ronald Brown