Fibration of simplicial commutative monoids

Question

Let $f:X\to Y$ be a homomorphism of simplicial commutative monoids, which is a weak equivalence (as homomorphism of simplicial sets) and surjective in each degree. Is $f$ a (trivial) fibration (of simplicial sets)? If it helps, we can assume $Y$ constant (i.e., concentrated in degree 0).

Answer 1

I don't think so. Given a set $S$, let $T(S)=\{0\}\cup S\cup\{\infty\}$ be the commutative monoid with operation $0+s=s$ and $s+t=\infty$ for $s,t\neq 0$. This construction is natural in $S$, so gives a functor from simplicial sets to simplicial commutative monoids.

Now let $A$ be a weakly contractible simplicial set which is not a Kan complex. Then $T(A)\rightarrow T(\Delta^0)$ is a surjective weak equivalence, but $T(A)=\Delta^0\coprod A\coprod \Delta^0$ is not Kan.