Kan replacement of finite $\mathbb{Q}$-type simplicial set

Question

Sorry if this question is maybe a bit basic, but it is on a rather specialized topic so I think it is more appropriate for MO than SE.

Suppose that $X$ is a simplicial set that has finitely many non-degenerate simplices in every simplicial dimension. Then of course $X$ has finite $\mathbb{Q}$-type (i.e. $H_n(X;\mathbb{Q})$ is finite dimensional as $\mathbb{Q}$-vector space for all $n\ge1$).

Let $Y$ be a fibrant resolution of $X$, that is, $Y$ is a Kan complex and we have a weak equivalence $X\stackrel{\sim}{\to}Y$. I would like to say that $Y$ also has finite $\mathbb{Q}$-type. Is this true?

This is where I get stuck: we know that being of finite $\mathbb{Q}$-type is equivalent to $\pi_n(Y;\mathbb{Q})$ is finite dimensional for all $n\ge2$ and $H_1(Y;\mathbb{Q})$ is finite dimensional. The first part of the condition holds because it holds for $X$ and $Y$ is weakly equivalent to $X$. However, I am unsure how to show that $H_1(Y;\mathbb{Q})$ is finite dimensional.

Any help and/or references would be greatly appreciated! I am also iterested in the case where we only assume $X$ to be of finite type (which should work with the same proof, hopefully).

Answer 1

Weak equivalences of simplicial sets induce isomorphisms on simplicial homology groups. In particular, the property of having finite-dimensional rational homology groups is preserved under weak equivalences.

The easiest way to prove this from scratch is the observe that the free simplicial module functor (in this case, we use modules over rational numbers) is a left Quillen functor from the Kan–Quillen model structure on simplicial sets to the projective model structure on simplicial modules because the right adjoint preserves fibrations and acyclic fibrations. In particular, the free simplicial module functor preserves weak equivalences because all simplicial sets are cofibrant. Thus, weak equivalences of simplicial sets are sent to weak equivalences of simplicial modules, which after passing to normalized chains become quasi-isomorphisms of chain complexes, hence induce isomorphisms on homology groups.