Query on w.e.'s

Query on w.e.'s

[categories]

Date: Fri, 31 Jan 1997 17:19:24 +0000 (GMT)
From: T.Porter <t.porter@bangor.ac.uk>

Dear All,

Can someone give a full reply to this? I was unable to give an adequate 
one with references etc as I have not been working in this area 
recently.  

Thanking you,

Tim.
 
---------- Forwarded message ----------
Date: Fri, 31 Jan 1997 15:21:48 +0000 (GMT)
From: Takis Psarogiannakopoulos <takis@dpmms.cam.ac.uk>
To: t.porter@bangor.ac.uk

Dear Friend
I am sorry that I disturb you with this letter but
since nobody here in Cambridge doent seem to know
an answer to something (and since I have seen your
name in Pursuing Stacks of Grothndieck) I am writing
to you with the hope that you probably be able to
answer me . In fact what I try to find out if it is
true, is a very "easy thing": In the paper of Quillen
Higher Algebraic K -Theory there is his theorem about
whether a morphism F:C--->B in Cat is a w.e.: if
we know that all comma categories F/b are contratible
(for any b in B) then F is a w.e. I am wondering if the
converse is definitely true , ie: if F:C---->B is a w.e.
in Cat (in the sense of Nerve functor) then for every b in B
all the comma categories are contractible.
In fact what I want to know is: if we have a commutative
diagram in Cat as
                   H : C -----> B
                       |        |
                    f  |        | g
                       J   =    J
(ie categories over J) where the map H is a w.e. (ie Ner(H) is
a w.e.  of simplicial sets) then for every object j of
the category J , the "map over j" H/j: f/j ----> g/j is a w.e.
Is something like that true? Since the idea of Ouillen in his
criterion is that "F/b plays the role of homotopy fibre of the
corresponding maps of classifying spaces" it seems to me that
the above is true.
But the fact that Quillen doesnt refer this explicitly to his
paper makes me wondering if there is a simple counterexample
where this fails (so there is no reason to sit down and try
to write a proof).
( I know that in the case that we define w.e s in Cat through
cohomology (ie restricting the w.e.s to the comological ones)
the above is true because we actually thinking with the corre-
sponding toposes but is this fact remain true for the case of
Nerve-w.e s ?)
I thank you in advance that you took the time and read this.

Sincerely
Takis
Department of Pure Maths ,Cambridge