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