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This may not be precise enough for MO, but I'll give it a go.

Let $M$ be a symmetric closed monoidal model category with unit $u\in Ob(M)$. We define the vertices of an object $A$ to be points $x\in \operatorname{Hom}_M(u,A)$. We define the fibre of $f:B\to A$ over a vertex of $x$ of $A$ to be the pullback of $f$ by the map $x:u\to A$ classifying $x$.

It can be shown in the category of simplicial sets with the cartesian monoidal structure (and unit $\Delta^0$) that a Kan fibration is acyclic if and only if all of its fibres are contractible (I'm pretty sure that a similar statement holds for CGWH spaces in the Quillen model structure). However, the proof uses specific properties of the category of simplicial sets to prove this fact. Is there some general additional structure on $M$ that we would need to prove this statement in more generality?

Edit: It's at least clear that we must require the monoidal unit to be trivially fibrant.

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In which general situation do you want to apply this result? –  Martin Brandenburg Sep 14 '10 at 6:29
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Or you could just ask: in a model category when is the following true of some class $S$ of objects? A fibration $X\to Y$ must be a weak equivalence if for every $s\in S$ the resulting morphism $s\times_Y X\to s$ is a weak equivalence. In this form it has nothing to do with monoidal structure or with $s$ being equivalent to the final object. –  Tom Goodwillie Sep 14 '10 at 13:36

1 Answer 1

up vote 3 down vote accepted

Consider sSet×sSet with the induced structure from sSet. The monoidal unit is (1,1) (which is trivially fibrant, as is the unit in any cartesian monoidal model category), and so an object of the form (X,0) has no vertices at all (with your definition). Hence any map (X,0)→(Y,0) has (vacuously) all its fibers contractible, but is not in general acyclic.

I think what you'd need is some sort of "well-pointedness" of M, which will probably be a very special property in general.

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Can you think roughly of a way to formulate this well-pointedness property? –  Harry Gindi Sep 14 '10 at 6:53
    
It would probably suffice to ask that the mapping space functor Hom(u,-) : M → sSet reflects weak equivalences. –  Mike Shulman Sep 14 '10 at 17:23

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