I've been thinking about the following question, and I was wondering if anyone else has given it some thought: Is there a good notion of an Sd/Ex adjunction for sSet/S equipped with the contravariant model structure (cofibrations are monomorphisms and fibrant objects are right fibrations over S) for an arbitrary simplicial set S? (Note: This is in the _unmarked_ case.) It seems to me that any sort of naive way of doing this (for instance, by pulling back the results in sSet (that is, given an object p:X->S of sSet/S, let Ex_S(p):=S\times_{Ex(S)} Ex(X) -> S with morphisms determined by the universal property)) is doomed to fail, since it does not incorporate the asymmetry of the model structure (that is, if that worked, it would also work for the covariant model structure, which seems like it shouldn't be true). One problem with trying to mimic the classical argument is that the classical/Quillen/Kan homotopy structure (this comprises the data of the model structure on sSet and all of its relativizations sSet/S for every simplicial set S (see Cisinski's book _Les Prefaisceaux comme modeles des types d'homotopie_ ch 1.3 for a precise definition, as well as some relevant results)) has the property of _completeness_, which is essentially the property that the weak equivalences of sSet/S are exactly the morphisms that map to weak equivalences under the canonical projection functor sSet/S -> sSet. Since the contravariant homotopy structure does not have this property, it seems imprudent to expect to be able to pull back results from "deeper" bases naively. Any ideas on how to come up with such a functor? Yours Cordially, Harry Gindi [For admin and other information see: http://www.mta.ca/~cat-dist/ ]