Kan's Ex functor for the contravariant homotopy structure

Kan's Ex functor for the contravariant homotopy structure

[Harry Gindi]

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


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