# In what generality does the following statement hold: A fibration is acyclic if and only if all fibres are contractible fibrant objects.

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
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