# Agal Algebraic Geometry – Do infinite products oscillate with trivial cofibrations to obtain simple quantities?

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!