Proper model category of simplicial rings revisited

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?

• I don't understand your claim that fibrancy of simplicial commutative rings uses the Dold-Kan correspondence. Every simplicial group is a Kan complex, and certainly that doesn't use Dold-Kan because it's true even for non-abelian groups... – Dylan Wilson Sep 30 '17 at 14:30
• And yes you can reduce to checking something on generating cofibrations: the collection of flat maps (which are just a little stronger than what you call an 'h-cofibration') is weakly saturated (i.e. closed under transfinite composition, pushouts, and retracts), so if the generating things are flat then everything is. – Dylan Wilson Sep 30 '17 at 14:46
• (A 'flat map' $A \to X$' (a notion I think is due to Hopkins) is the following: for every $A \to B$ and weak equivalence $B \to B'$, we ask that $X \coprod_AB \to X \coprod_AB'$ be a weak equivalence.) – Dylan Wilson Sep 30 '17 at 14:52
• Oh whoops, I misread the definition of your 'h-cofibration'- I think it's the same as 'flat'. But anyway, the main point remains. – Dylan Wilson Sep 30 '17 at 14:54

This paper 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.

• This is excellent. Perhaps it is possible to see that $\pi_*P[x]\simeq \bigoplus \pi_*P$, naturally in $P$, without invoking the isomorphism of graded abelian groups $\pi_*P[x]\simeq H_*C(P[x])$? The condition in Thm. 9.1 on $(\cdot)\otimes_\mathbf{Z}\mathbf{Z}[x]$ would be met, and properness would follow. I haven't given it a thought yet. – user97068 Sep 30 '17 at 21:10

This is just a small complement to Charles' answer. The category of simplicial abelian groups is finitely generated (see Hovey's book $\S$7, for instance) and domains and codomains of generating cofibrations, respectively trivial cofibrations, are small relative to the whole category, since they are finitely presented $\mathbf{Z}$-modules. Running the same proof of his Lemma 7.4.1 with no need to assume the transition maps in his statement to be cofibrations (because domains and codomains of the generating cofibrations/trivial cofibrations are small with respect to the whole category, and not just with respect to cofibrations), you obtain that, in simplicial abelian groups, filtered colimits preserve fibrations and trivial fibrations. Since they already preserve cofibrations and trivial cofibrations, they preserve weak equivalences. In particular, a coproduct of weak equivalences is a weak equivalence, and you can use this in place of the graded ring isomorphism $\pi_*P[x]\simeq H_*C(P[x])$ to conclude that $(\cdot)\otimes_{\mathbf{Z}}\mathbf{Z}[x]$ preserves weak equivalences of simplicial rings.

This small addendum to the very last paragraph of the answer, together with the criterion that was pointed out in the answer, gives the proof you were after: no use of the Dold-Kan correspondence, not even in the form of the identification between homotopy groups and homology groups.