Let $s\text{Ring}$ denote the category of simplicial commutative rings.
We endow it with the model structure defined by declaring that fibrations, trivial fibrations and weak equivalences are, respectively, those maps inducing fibrations, trivial fibrations and weak equivalences on underlying simplicial sets.


I'm interested in finding a proof of left properness of $s\text{Ring}$ that does **not** use the Dold-Kan equivalence, either directly or indirectly. In particular, without using the fact that simplicial rings admit a forgetful functor to simplicial abelian groups.


*[The proof that I can think of using the fact that every simplicial commutative ring is fibrant (hence, implicitly, the Dold-Kan equivalence) goes this way:
let $A\to B$ be a cofibration in $s\text{Ring}$, and $A\to C$ a weak equivalence. The pushout $B\otimes\_AC$ (degree-wise tensor product) is equivalent to the derived tensor product, as $B$ is cofibrant as an $A$-module (Corollary on page 6.10, Ch. 2 of Quillen's "Homotopical Algebra").
Now by Thm. 6(b) in loc. cit., we deduce the map $B\to B\otimes\_AC$ induces an isomorphism on $\pi\_i$ for all $i\ge 0$, and hence is a weak equivalence. QED]*


However, the above proof makes use of the fact that every simplicial ring is, in particular, a simplicial abelian group, in a crucial way, e.g.. in applying Thm. 6(b) quoted above, or the corollary quoted right before.


I would like to find a proof that, rather, shows that every free morphism is an $h$-cofibration (in the sense that pushout along it preserves weak equivalences). For the definition of free morphisms in simplicial categories, see Goerss' notes on simplicial methods, Def. 4.19 and the discussion around it.


Every cofibration is a retract of a free morphism (and the converse is also true), and $h$-cofibrations are stable under retracts, hence it's enough to show free morphisms are $h$-cofibrations.


Can one reduce to the case the free morphism in question is a generating cofibration, ie. of the form $\mathbf{Z}[\partial\Delta[n]]\to\mathbf{Z}[\Delta[n]]$, $n\ge 0$?


Can one reduce to showing that for every cofibrant simplicial ring $A$, the functor $(\cdot)\otimes\_{\mathbf{Z}}A$ preserves weak equivalences?


Is anybody aware of such proof, or of general left properness criteria that might just apply?



Proper model category of simplicial rings revisited
[This paper](https://arxiv.org/abs/math/0003065) proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly includes the case of simplicial objects $s\mathcal{A}$ in a category $\mathcal{A}$ of algebras which can be described by an algebraic theory. In particular, simplicial commutative rings. (The paper tries to be general and deals with "many-sorted" theories, but I'll just talk about the case of a single-sorted theory: i.e., objects of $\mathcal{A}$ are "sets with algebraic structure".)


In any case, the theorem proved there (Theorem 9.1) says that $s\mathcal{A}$ (with the model structure defined by Quillen) is left proper if and only if the functor 


$$P\mapsto P\amalg F \colon s\mathcal{A}\to s\mathcal{A}$$


preserves arbitrary weak equivalences. 
Here $F$ is the constant simplicial object on the free algebra on one generator in $\mathcal{A}$.


So for simplicial commutative rings, we just need to check that the endofunctor $P\mapsto P\amalg F \approx P\otimes \mathbb{Z}[x]\approx P[x]$ on the category of simplicial rings preserves weak equivalences. This is clear: as a simplicial abelian group, $P[x]\approx \bigoplus\_{n\geq0} P$. 


The only fact about simplicial abelian groups $A$ that you need here is that $\pi\_\*A\approx H\_\*C(A)$, where $C(A)=(\dots \to A\_2\to A\_1\to A\_0)$ is the associated chain complex, so that you have $\pi\_\*P[x]\approx \bigoplus \pi\_\*P$. The general fact quoted above makes no use of simplicial abelian groups, and objects of $\mathcal{A}$ might not even have an underlying group; e.g., the theorem applies in the case $\mathcal{A}=$ associative monoids. 


