I read the book by Voevodsky and Morel$ mathbb {A} ^ 1 $Homotopy Theory of Schemes & # 39 ;. In Remark 3.1.15 this is stated for every simple fiber color $ F $ and open sentences $ U subseteq V $. $ F (V) to F (U) $ is a vibration.

Prove by definition. We have a bifunktor

$$ begin {array} {ccccc} sSet & times & Shv (Sm / k) & to & sShv (Sm / k) \ (S &, & F) & mapsto & S times F end {array}, $$

from where $ (S times F) (X) _n = S_n times F (X) $, Consider the coequalizer

$$ lambda ^ n_k times U rightarrows lambda ^ n_k times V coprod triangle n times U to C. $$

Then there is a card $ i: C to triangle n times V $ and the question is reduced to the RLP of $ F $ w.r.t $ i $, So I want to prove it $ i $ is a trivial co-calibration.

It's obviously a co-calibration, but I'm determined to prove it's a weak equivalence. Suffice it to prove that the functor $ – times F: sSet to sShv (Sm / k) $ is a left-hand quill functor because we could then use the pushout diagram of $ C $, So we will prove that trivial cofibrations are transformed into infinite products by transferring them to stems …

I think we have to prove that geometric realization functor commutes with infinite products, at least to a weak equivalence. Is that true?

Thanks a lot!