Left Proper model structure on the category of non-symmetric operads in chain complexes

Question

It is shown in Moriya (Multiplicative formality of operads and Sinha’s spectral sequence for long knots, 2.1) that there exists a left proper model category structure on non-symmetric operads over $k$-chain complexes $\textrm{Ch}(k)$, where $k$ is a field. He gives a direct proof of the theorem, and I am not sure whether the hypothesis of $k$ field is used somewhere.

I am trying to understand if this holds for $k= \mathbb{Z}$ too. In Muro (Homotopy Theory of non-symmetric operads), a model structure is shown to exist in much greater generality, and it is shown that if $\textrm{Ch}(k)$ is combinatorial, then so is the model category on $\textrm{Ch}(k)$ non-symmetric operads. However, left properness is not treated, and I can't figure out in an easy way what the generating cofibrations should b (to prove myself lef-properness).

EDIT: all chain complexes here are meant to be non negatively graded.

Answer 1

Let $k$ be a commutative ring and let $\mathrm{Ch}(k)$ be the category of non-negatively graded chain complexes of $k$-modules. We endow it with the projective model structure. Weak equivalences are quasi-isomorphisms and fibrations are maps which are surjective in positive degrees. As a consequence of Theorem 1.11 in…

Muro, Fernando. “Homotopy Theory of Non-Symmetric Operads, II: Change of Base Category and Left Properness.” Algebraic & Geometric Topology 14, no. 1 (2014): 229–81. https://doi.org/10.2140/agt.2014.14.229.

… we obtain the following corollary:

Corollary: If all objects in $\mathrm{Ch}(k)$ are cofibrant then the category of non-symmetric operads in $\mathrm{Ch}(k)$, with weak equivalences and fibrations defined arity-wise, is left proper.

This sufficient condition holds for $k$ a field, but not for $\mathbb{Z}$.

It is easy to see that the category of non-symmetric operads can’t be left proper if there exists a $k$ module $M$ and $i>0$ such that $\mathrm{Tor}^k_i(M,M)\neq 0$ (this rules out $\mathbb{Z}$). Indeed, let $f\colon P\to M$ be a projective resolution. We can regard $f$ as a morphism of operads concentrated in arity $0$ (the arity $1$ part should be the identity in $k$ because of the operatic unit, but the rest is $0$). Then, if we freely add an arity $2$ generator to the source and target of $f$, we obtain a direct sum of tensor powers of $f$, including $f\otimes f\colon P\otimes P\to M\otimes M$, which is a quasi-isomorphism iff $\mathrm{Tor}^k_i(M,M)=0$ for all $i>0$. Notice that this doesn’t exclude commutative Von Neumann regular rings which are not fields.

In general, by the aforementioned theorem, we have a more general result, very close to left properness, which is sometimes enough for applications.

Theorem: A push-out of a weak equivalence of non-symmetric operads with underlying cofibrant complexes along a cofibration of operads is always a weak equivalence.

Notice that, over $\mathbb{Z}$, a complex is cofibrant if it consists of free abelian groups. This is the case for operads arising as singular or cellular chains on some geometric object.

The previously cited paper is a continuation of…

Muro, Fernando. “Homotopy Theory of Nonsymmetric Operads.” Algebraic & Geometric Topology 11, no. 3 (2011): 1541–99. https://doi.org/10.2140/agt.2011.11.1541.

These papers deal with both operads and algebras. I made a mistake in the algebra part of the first paper, which spread to the second paper. This doesn’t affect the operad part, which is what you’re interested in, but just in case let me copy the reference to the erratum:

Muro, Fernando. “Correction to the Articles ‘Homotopy Theory of Nonsymmetric Operads’, I-II.” Algebraic & Geometric Topology 17, no. 6 (2017): 2–3852. https://doi.org/10.2140/agt.2017.17.3837.

Hope this helps!